Look, if its the coil packs they aren't going to all fail at exactly the same time, I've just had a coil pack problem on my XKR, the symptoms were just a slight misfire that I couldn't get rid of by plug ,leads etc. The very fact that yours starts straight away is a good indication that coil packs are fine & valve timing is fine.
Its difficult to try & diagnose from 400 odd miles away, but the way it seems to me is that after it fires up & then stops straight away then you are losing power to either the injectors or the fuel pump, this is where the 12v path alters from where it is directed from ignition on (engine stationary), start cycle same path, engine fires, key released electrical path changes via relays to run condition. If on releasing key engine then ceases to run then you must be either losing power to pump or injectors or seriously over fueling, I cant see that its running long enough from your description for it to be the cold start valve or thermal switch so I come back to wiring / relays. I think the solution will be simple in the end. By the way we are talking 4cyl here ??? If I was nearer I would come & have a look & maybe consider putting a constant 12v supply to the fuel pump from the battery & see if it continues to run. If it stops the its not the fuel pump. I got your mail just now & I think I've answered it as far as I can at the moment, but I'll keep it ticking over & see if I can think of anything else.