[SimonP]

Hi!
The lift pump is located inside the fuel tank.
The access to it is from a removable panel in the boot floor, between the spare wheel well and the rear seats.
Before removing the pump/sender assembly from the tank, test for voltage at the pump:

If you pull back the rubber boot from the electrical connector, you will see a green/purple wire, which is thicker than the others. This is the power for the pump.

The fuel pump is fed through fuse 23 under the bonnet on E34 models.

The relay for the fuel pump is energised under TWO conditions:
1: Terminal 86:
A feed is tapped off the glow plug relay output to cylinder 6. This allows the lift pump to prime the fuel system from a cold start.
This supply is present when the glow plugs are being operated (the glow plug light is on), and during "afterglow" - the glow plugs may be intermittently energised after starting, to aid the clean running of a cold engine)
2: Terminal 86b:
A feed is taken from alternator terminal "D+". This is a safety feature - if the engine stops rotating, the alternator stops producing output and the fuel pump relay will be de-energised.
You should have this input whenever the charge warning lamp is NOT lit with the ignition on.

My guess is that your fuel pump relay is not responding to the alternator, therefore once the afterglow phase has finished, the fuel pump will not run, and your engine will stall.
Once the engine temp is outside the blue zone, the glow plugs are not activated, so this sounds like why the car will re-start if left to cool.

The fuel pump relay is a special-purpose relay - it has two diodes in it so that it can be energised from any of the two sources as above. A standard relay is not suitable.

Make up a test strap from a piece of wire with a 1/4 inch male blade each end, to mimic the relay pins.
Remove the fuel pump relay and insert this test strap between the sockets which correspond to terminals 30 and 87 on the relay.
Start the engine and see if it will stay running.
If it does, you need to check for voltage on the relay socket which corresponds to relay pin 86b.
If you have this, the fuel pump relay is faulty. You may however just be able to replace the diodes, as the relay appears to be working from the glow plug input. It may even just be a poor solder joint that wants re-making.
If you do not have voltage on terminal 86b with the engine running, there is a disconnection in the wiring loom.

Simon.