Check the battery for a voltage reading circa 12.6 volts is a good fully charged battery.
Next get the battery a drop test to check its giving enough amps,you seem to have two faults so low system voltage may be to blame.
The glow plug relay powers the lift pump when the glow plugs are on.It also powers the lift pump in normal running after it receives a live signal from the alternator once the engine is running,hence the bad hot starts because no lift pump until engines running.
So from the above you can see the relay has two sides or functions to it,now running a live to the lift pump without first disconnecting the original live wire to it will fire the glow plugs,not sure if they stay on or just run the normal cycle but they will come on.
It may be that the glow plug relay does other functions and sends power to engine sensors,I am not sure of this but if it does this may explain the injector warning light coming on and the limp home mode.
I would suspect that the glow plug relay is not getting a live signal (ignition barrel ?) or it has packed up ? and if when you put a live to the lift pump (if the original feed is still connected) that this maybe powers the relay up and causes the ecu to light the warning lamp ?.
Other scenarios are water in the ecu compartment,check for a short on any wiring in the engine compartment or loose connections,and esp boot/tailgate areas that may cause the whole system to short out.
There is not much in the way of info on the net concerning these Bmw tds's so have a look on some Land river forums as the discovery p38 shares the bmw engine and you amy pick up a few pointers there.But I would presume the glow plug relay has other functions and is not supplying power elsewhere to something and the ecu not receiving a signal flags the injector light and goes limp mode.The glow plug light staying on points to a problem with it.
I have no idea where the glow plug relay is on these cars ?,oh check the engine earth lead as well (drivers side front of engine level with sump pan).
