2001 V8 Discovery starting problems fuel / Immobiliser ??

My 2001 has had intermittent starting problems. Normally she starts first turn of the key.
I had 2 new keys cut and since then I have had a problem with starting the car. I thought it may be the immobiliser but the garage says if it was the immobiliser, then the car would not turn over and the red light on the dash would not go out.

The car can start first time but after running for a few miles and I stop and turn the engine off, the engine turns over but will not catch and start. If I leave for 10 – 20 mins and try starting again she starts first turn of the key.

They say it is the fuel pump. If it is the fuel pump, I live in Northampton, where is the best place to buy one, which model number / part numbers do I order and how easy are they to fit. I am fed up of being ripped off by over charging garages.

not the immobiliser.
with the ignition on-not cranking engine can you here the pump humming? if not check the fuel pump relay first. If thats ok then yeah it could be pump...take the fuel filter off & put the ignition on-no diesel=no pump.
Get your fuel pressure reg checked too, needs to be 4 bar pressure at the engine.
Theyre the do it yourself possibilities...

If you still dont have any joy, get it plugged in-could be electrical