LEDs in indicators

The OP has stated the lamps work properly when used as the indicators, ie Flasher unit not the issue, BUT when supplied with current from the vehicle it's flashers are working but the trailer isn't working properly. The current to supply each trailer lamp is coming through the same wires.
If the vehicle works ok, then anything connected to the vehicle is additional load and if anything, may make the flashers run slower.

Only if you lighten the electrical load does the flasher unit run fast.
Load resistors and the like isn't the problem as far as I can see.

Often with semi trailers the lamps are multi voltage and when fed 24v the LED lights see, first 12v and then 24v and they change in brightness. If the total load of the circuit is a bit much for the connections in the systems, ie, resistance, and the applied voltage to the LEDS keeps changing and that makes them vary

I have some truck LEDS white, orange and red stop/tail. They are buggers of things to make work without the brightness varying as it jumps from 12 to 24 mode. However, the OP hasn't got truck lamps, just replacement globes.

I would still look at the tow vehicle - older vehicles simply connected the left and right turn signal circuits together at the hazard light switch - newer cars aren't quite so simple with all of this being chip controlled now. It is probably a different program that runs the hazards compared to the turn signals.

How often do you use hazard flashers anyway? Unless you vehicle is physically unable to be moved, there should rarely any need to use hazard warning lamps (other than when driving thick fog where they are permitted) - if your vehicle is in a position that is causing a hazard, you should move your vehicle to somewhere it won't be a hazard rather than just flicking on your warning lamps.

The left and right indicators work separately so I cant see the plug being wrong.

Ivan
But they can earth through the other side and appear to operate normally. When both positive supplies to the L&R indicators are fed via the hazards then the earthing via the other side is negated and the lights won’t operate.

They are LEDS. Light Emitting Diodes. Diodes only conduct one way. You cant conduct backwards

Ivan
We are not talking about back flow through a one way diode at all.
Not sure why you think that.
The lights on the trailer CAN find a return path through the wiring of trailer and the vehicle indicator system. There won't be current flowing backwards through the diodes at all. Simply power applied to a LED globe CAN, if it's earthing isn't correct, to find a path through the opposite side of the vehicle system. We don't actually know what is happening there yet. When both sides of the system are powered, ie, hazards are on. the power is applied to both sides and so one side or the other cannot find that path because there is the same polarity at both sides at the same time. It is a fault somewhere. Maybe in the plug.

Some systems rely on that happening to turn a light OFF. ie, ALt charge light on, engine stopped. Start engine, alt charges and same voltage appear across the indicator globe, which indicates it is charging. Some of those are LEDS too.

Ok I've now spent more time on this. At first I only fitted an LED globe to one indicator light. That was when that light didn't flash with the hazard flashers. When I fitted an LED into both indicators they both work as normal. The LEDs flicker slightly when operating but when indicating they are much brighter than normal globes. When the hazard flashers are used the LEDs are not as bright but serviceable and the increased brightness of the indicators make the change worthwhile. Thanks for all the comments.

We had the opposite problem - the caravan had two turn-signal lamps for each direction (one at the rear, one on the side near the front). While the Prado had no issue with the standard globes, the X-Trail flasher controller couldn't handle the load and the lamps would just stay on continuously. I replaced the front side turn signal lamps will LEDs to reduce the load and we can now tow the van with either vehicle.