Range Rover Vogue Stuck In Low Range

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Looking at this video bypassing the switch is not so easy as you will need to identify the correct wires.
You could probably test the switch if you have a multimeter by putting the probes on the wires.

No lights on dash to indicate in low range

If it's properly in low range you should have this on the dash

Same thing happened to me on my first L322, the shift motor was stuck, took it off and stripped it on the bench... juiced it back and fore with 12v a few times before having the headache of trying to work out where it needed to be timed to before refitting.

Exercised it a couple of times a week after that.

Switch is also prone to suffer from drink spillages as the cup holder is just aft of it.
Friend got under her today and it’s the ratio motor that’s gone taken it apart and put back together and now won’t select gears at all
This is from the manual and may be relevent given yours was stuck in low range.

If the actuator and gearbox assembly is to be removed, the synchroniser sleeve position must be set to high range
using TestBook/T4 before the assembly is removed.