The Fast and the Furiously Crazy

Since you've eliminated the sensible solution (bypasses or double tracks), let's go with an insane one!

All your trains have rail tracks running on top of them and extra wagons with ramps at the front and back. When a priority train approaches, they lower the ramps onto the rails and the priority train drives straight over them as if it were a bridge.

Some notes:

• The slower train should drive at maximum speed if overtaken from behind or stop if the prio train approaches from the front.


• The ramps will need to be very long to prevent the prio train from jumping the tracks or going completely airborne. Rollercoaster-like guardrails could assist here, adding the weight of the bottom train to the downward force.


• This only works with very straight tracks. Very. Straight.


• It is not recommended to attempt overtaking a train that is overtaking another.

A more boring but slightly more sensible variant of this is that the slower train stops at a depression/valley in the track so that the rails on top become level. You could even have moving sections of track that can lower so that the slow train when stopped there is essentially replacing that section with its roof rails. This might count as a fixed bypass though.

Sideways treads

Here is a second more boring solution: All locomotives and wagons are powered and carry retractable treads underneath that allow sideways movement. These treads are wide and solid to support the train, but have indentations to protect the rail bars from damage.

When a train is commanded to make way it comes to a stop, lowers the treads to the ground and moves to the side until it is clear of the tracks. After the prio train has passed, it rolls back onto the track, aligning the wheels with the rails carefully, then retracts the treads and resumes its journey on the rails.

In these trains, each wagon has its own electric motors for driving both wheels and treads. This increases the total weight of the train, but distributes it better than pure locomotive/unpowered wagons so there is no 250 ton locomotive to move onto the mud. The locomotive in this case mostly houses the (diesel) generators that supply electricity to the train and the controls.

The ground next to the train tracks needs to be level and sturdy enough to support the weight of the train, but not quite to the standard of the rails themselves.

Two trains become one.

On the track there is a train heading in one direction at 100 km/h and a second train heading towards it at 200 km/h.

You do not specify the distance between them. If there is some distance there is time for this maneuver.

1. Slow train slows down, stops, goes into reverse.




2. Previously slow train accelerates, in reverse, until it is going almost 200 km/h.




3. Fast train will slowly close the distance. When the two trains are very close, they are linked. This is not something routinely done with fast moving trains but is done all the time with slow moving trains. It does not seem outrageous; relative to one another the trains are barely moving, or even not moving. It is akin to refuelling a plane in flight except easier because the trains are on the same track.




4. You now have one fast train, going the specified direction at the specified speed. The fast train did not have to break stride. The slow train did not leave the tracks. You did not have to build anything new.

The neat thing about trains is that the size of the train is fluid - it can be longer or shorter according to need, and trains can be merged and split.

I have an idea which I'll try to put into writing but it might not be obvious what I'm driving at. I actually had a couple of thoughts on this but one may be more sensible than the other.

Up and Over

My initial thought was that one of the trains, probably the 200 km/h one as it would already likely be a streamliner, would be designed in such a way that it's front is like a ramp and it has rails built into the ramp which run along the track in front of it. These rails continue over the carriage roof and the rear of the train look like the front. The oncoming train could then be forced to run over the top of the fast train. This does however require that the train that goes over is able to climb an unrealistic gradient although oncoming speed may assist. It also assumed that there are no overhead electricity cables but that the train above is able to continue to proceed without a "third rail" either.

Shall we dance?

As an alternative to the up and over method, I came up with something that could actually work. On a traditional track, you'd place the 100 km/h train in a siding and have it wait until the 200 km/h train had passed. I realize that this can't work because it requires you to know where the siding would need to be.

However, when you consider what a train requires, it is essentially rails on which to run. Now, imagine that each train is carrying some sort of short section of rail at its front which is angled from the right to the left of the track. When the two trains come close enough, these angled tracks collide and are forced into the rail bed. This causes the left-hand wheel set of each train to jump off the left hand track and the right hand wheel set is forced onto the left hand rail. This effectively derails both trains simultaneously causing a massive accident. However, if the tops of each train were designed to carry some sort of rail / tube on top of the train with some sort of interlocking arm, the weight of each train would be carried by the other.

Each train would hold the other up, a little like a spinning ballerina is supported by one of two feet and by a dance partner at the top of their arm, outstretched above their heads. Both locomotives would progress along the same piece of track but using just one rail each, each locomotive offset and supported by the other.

Once they had passed each other, an assembly at the rear of each train could "re-rail" the wheel sets back to their original location which would effectively by a mirror of the assembly at the front of the train.

Hopefully this makes some sense. If not, I could possibly try to sketch out how it would look.

Edit Added sketch

Edit 2 — "Budge Over" trains

You could potentially re-design the trains to allow the "shall we dance" method to be a little less severe. Instead of forcing the trains to jump to the opposite track, design the trains to have an angled cab with looks like a triangle when viewed from above. Along one side of the train, have interlocking "rails" which would interface with each other, shoving the oncoming train onto the side of the opposing carriage.

Redesign the wheel sets so that they had one fixed set of wheels and one "sprung" set which were floating so the trains could continue to run one wheelset on a single rail but the other would float in free air under the train.

This would effectively allow trains to "slither" past each other. Whilst this refinement to the design probably makes the solution a little more realistic, it does remove some of the grand drama that the "Shall we dance" method has.

You can't

Well you can, but it's going to take a few days. First you're going to need to get the infrastructure in place.

A train weighs between 1500 and 6000tons. I'm assuming this is a passenger train rather than a cargo train which could weigh nearly 100,000 tons.

We're going to need cranes in place able to lift at least 250tons just to move the engine out of the way. Such things do exist as they're part of the breakdown and derailment recovery processes but they're not exactly common, it may take a day or two just to get them into place.

You also need to make sure there's a safe and stable surface to put your engine onto that isn't the tracks. It still weighs up to 250tons, so you can't just put it down on unprepared ground and preparing ground for that sort of load takes time.

I'm sure you can see where this is going. There's no quick and safe way to remove a train from the tracks.

Accidental derailment also damages the tracks often over long distances, a general theme is that it takes a week to ten days to recover and repair after a relatively minor incident.

Jackscrews

The slow train will come to a complete stop. At both ends of each of its cars, two outrigger hydraulic- or screw jacks (each as tall as the train) are extended outwards to beyond the fast train's width, and after this downwards to beyond the height of the fast train, lifting up the entire slow train. Effectively, this forms a tunnel underneath the slow train through which the fast train can travel.

After the fast train has passed, the train will be lowered exactly to its original position. If necessary, minor corrections can be performed laterally by varying the outrigger's position.

For those who believe one cannot lift a heavy train with compact jack screws or hydraulic jacks, look no further than self driving cranes.

Apparently, a prototype already exists:

(both images from Wikimedia)

Use road wheels on the slower train. Build it out of railcars like these:

When the faster train approaches, just stop, raise your rail wheels, and drive off the tracks.

On the slow train by each set of wheels also put a set of 3 wheels at 90 degrees to the main ones, each on an extending pistons.

When it needs to let the fast train pass it stops and lowers the wheels on pistons, then drives sideways off the track (it needs 3 so it can lift one to pass over the rails and still have 2 on the ground).

Once the fast train has passed it reverses the process to drive back onto the train and recenter itself then lifts the extra wheels and continues.

Depending on your technology level, all train tracks could be raised off the ground and have rails above and below. All East/North bound trains run on the top rails. All West/South bound trains travel hanging on the bottom side of the tracks. Inside the cars are tubes so that the ceiling can always remain up. Every 10 kilometers or so, you could have loops that could move a train from one track to another, so a faster train could overtake and pass a slower train going the same direction.

Your only real hope is a track gang who can move fast.

They find a place along the track near the starting position of the slow train, where a bolted joint exists that's staggered by just a few feet between the two rails. It should not be a special joint such as an insulated joint. If it's not there, get a rail saw and rail drill in advance, and create a joint. The crew stages out there with a bulldozer and a number of semiloads of track panels and a crane, and they lay down enough temporary track to easily fit the slow train.

Long before it arrives, they unbolt the main track, shove one main about 5 feet to the left, the other 5 feet to the right (the staggering decides which goes which way), grade the subgrade to level, and drop in the temporary track to meet it. Throw 2 bolts in the joint bars, no more. Have the slow train crawl into this temporary track, and about 100' past the joint.

Now you have 15 minutes to reverse. A bulldozer is already chained to the temporary track, and six other bulldozer or big SUV winches are tied to the main track, ready to pull the segments back where they belong. Yank the 2 bolts, pull the tracks over, and a few workers tighten the 6 mainline bolts while many other workers with gas powered jackhammers tamp the main track back to level. ZOOM, the other train tears through.

This is achievable with a crew that knows what it's doing. Railroad track is "lego" like that.

Rinse, wash, repeat to back the first train out onto the main again, reassemble the main, and the first train is on its way.

Of course you know, nobody goes 200kph without some sort of automatic signal systems to prevent collisions. There'll never be any danger of collision, because the signal system will stop the trains if a train is in the way or the track is severed.

Hold the regular train at the last switch or junction

The question says:

Answers adjusting the infrastructure or train construction before the encounter are acceptable, so long as there are no fixed bypasses or signals.

Rail roads always have sidings and some form of communications specifically to keep trains from arguing over right-of-way while at mutual approach, better phrased as "colliding". Therefore, by construction of the question, at each end of this long section of single track there is a switch for double tracking, siding, or some other place where a train could wait.

A high priority train in the absence of traffic signalling should run on a timetable. The rail traffic control point at each end of the long section of track will have the latest timetable for when the high priority train is scheduled to enter and to leave that section. They will therefore hold all oncoming traffic until they observe that the high priority train has departed the section.

The rail road company would probably make a major effort to improve signalling between the control points, so that they can communicate when a train enters the section and when it leaves. This might be beyond the scope of your work, however.

This infrastructure isn't only for the benefit of the high priority train -- this is for the benefit of all trains, and maintenance as well: Any track work or blockage has to be handled without additional trains making the situation more blocked.

Edit based on comment:
Even for an unscheduled express or emergency train -- especially for an emergency train -- the rail road will have some form of communication specifically to prevent unintended cases of head-on collisions.

No problem. Captain Kirk is orbiting above the trains in the Starship Enterprise. He has Scotty take over the transporter and Scotty beams the first train behind the second to avoid the collision.

An impractical but possible 2nd answer.

The fast train has lifts at both ends and tracks on the roof.
The slow train stops, reverses, and allows fast train to catch up.
Fast train now uses front lift to raise carriage, roll it across the top of itself and put it back down with the opposite lift.
Does this with all carriages and engine then the 2 seperate and the slow train stops, reverses, continues.

Use a large Outrigger Suspension system

If you google "Outrigger on crane" you will find many examples of retractable rigging on smaller vehicles to prevent them from tipping over, usually cranes. The trains in your world could all be fitted with a variant of this system that is designed to hydraulically lift the cars of the train high enough to allow the other one to pass beneath.

The contact points would all be well outside of the track and it would not matter much the location at which the train stops since each contact point can be raised to a variable height. The best part is that it would only take a couple minutes to fully suspend the train. Also, since all trains could theoretically be outfitted with something like this, shifting priorities in the train schedule would have little impact.

Furthermore, the use of this technology could be justified by explaining that the cost of implementing something like this on all trains is cheaper that laying a completely separate parallel track.

Add a tilting mechanism to the carriage wheels and the side of the slow train, allowing it to stop, safely tilt itself off the side of the track, and then re-stand itself onto the track after the high-priority train passes safely.

Step 1: The train stops.
Step 2: The train extends the side mechanism all the way to the ground.
Step 3: The train extends the wheel-side mechanism while retracting the side mechanism, such that it rotates 90 degrees off the track onto its side.
Step 4: The train retracts the wheel-side mechanism, leaving the train laying safely on its side.
Step 5: The train uses the side-mechanism's articulation to pull itself a safe distance from the tracks so it out of the way of the priority express.

To re-rail the train, simply reverse the process.

The second train has a lower area for the first train to pass through.

You would have to use belts,gears and chain, or something to connect the second trains engine to the wheels.

Either that or the second train is 100% electric and multiple train cars have motors to spread out the work load.

The most realistic way to do it fast, if you don't have the infrastructure in place, is to:

1. Stop the less prioritary train;


2. Use a crane to move it off the rails;


3. Once the priority train has passed, use the crane to move the less prioritary train back onto the rails.

Cranes are awesomely strong. Just go to Google and do a dearch image for 'crane 100 tons'. You'll see that a lot.of relatively small models can lift that much. Also remember that even if your train weights much more than that, you only have to lift one wagon at a time.

Helicopters

Assuming money is no object--this will be expensive, and it will require precision high speed work by skilled personnel. It should result in no harm to equipment or passengers.
There are a number of models of heavy-lift helicopters out there; wikipedia has a list. Attach lifting cables, decouple the train cars, and lift them out of the way individually.

former train stops and reverses. Fast train catches up. Passengers swap over to opposite train. Fast train decouples, stops, reverses direction. the former train is now the fast train, and the passengers never slowed down.

Rockets

Given that there are long stretches of track that the high-priority train may be traveling on, the design of the high-priority train must take into account this scenario. Ideally, it is able to handle these situations itself, as the oncoming slow train might not have the means to get out of the way, the slow train's mechanisms might fail or be maintained to a lower standard than the high-priority train's operation safely permits, etc., and the slow train might not be capable of match the high-priority train's speed for coupling manoeuvres.

The high-priority train can take care of the situation itself if it elevates itself above the track for a brief period of time. Rockets can produce enough thrust to lift a train, although each engine and carriage will need rockets attached, and will need enough fuel to fire twice - once to lift, and once to land. This will be a significant portion of the train's weight. The force will need to be great enough to lift the train for long enough to pass the underlying train; at a relative speed of 300km/h it will take 24 seconds for both trains to pass each other, and (disclaimer: I don't know rocket physics) a set of rockets with ~450kN thrust should be able to lift the train to around 250m for a safe passage, and the fuel for this will be an extra 40 tons. You'll probably need more fuel (and slightly more thrust) in case the high-priority train needs to do this multiple times, the trains' combined length is longer than the 2km I used, and to ensure you have impulse to spare when landing to ensure you can line up with the tracks again accurately. These numbers are for a 250-ton engine; less-powerful rockets will be needed for the carriages depending on their weight.

Rocket engines of this size are similar to the Merlin engines in the Falcon; depending on the number of carriages the train has, it might have a comparable amount of thrust to the complete set of engines in a Falcon-9. It still won't go very high though, because the train is aerodynamically terrible going in a direction it's not meant for, and it won't be carrying much rocket fuel.

Magnetic Levitation

In a scenario where a fast train needs to cross a slower train on same track, it might make sense to levitate one of the trains above other.

Perhaps a levitation technology could be built into maglev trains, that allows one train to rise enough to let other pass under it on original track, without requiring other to slow down, and without pulling flying train down when other one passes under it.

Perhaps lower train could have maglev tracks on roof to support flying train when they are passing each other.

I suppose only two problems are facing us, looking at the current technology:

1. How to balance flying train: train may have tendency to fall to one or other side of track if levitated beyond a certain height. Will need support to balance above track.




2. Enormous power requirement: Through for a short duration, a lot more power than usual would be required to lift a train a few meters instead of a few mm.

PS: I don't know much about trains.

Catch Points

Catch Points exist explicitly for this scenario, to redirect a runaway train off a track or away from a dangerous situation without requiring it to be slowed down.

If you need a nice ad-hoc solution using no specific rail safety measures, perhaps a dead-end track-switch or branch that never went anywhere would work in place of a purpose-built catch-point.

Just remove the end-cap/bumper that blocks the track, switch the low-priority train onto that, drive straight off the end of the track and coast on solid ground.

Once the low-priority train is out of the way, switch the track back and watch the high-priority train rush past safely.

These dead-ends can be found in a lot of places, often in places where a train track has been torn up due to age or redundancy. They might be in documentation somewhere, but narratively it's the sort of information an expert on the area would have and someone who just travels the line wouldn't notice.

This approach would be great for dramatic effect.

Portals

Simple, place a blue portal in front of the fast train and an orange portal behind the slow train. Collision averted.

Of course, if you've got portal technology the need for trains might be a bit moot.