Well if you have a stock engine then there are no crankshaft or camshaft position sensors like a modern EFI car would have. And the water temperature sensor only tells the cold start relay when to fire the cold start injector solenoid in the throttle body when starting a cold engine. There is a whole host of simple things to check, but one that comes to mind is that start solenoid relay. When they go bad they stick "on" and will flood the engine after it starts. It is easy to check, just disconnect the wires to that starter injector to disable it, then start the car and see if the problem goes away. If not, it is something else.
A few initial questions:
How long will it idle before dying?
How do you know you actually have enough fuel pressure?
How long have you had the car, and was any work done on it just before this problem started?
Do you have a shop manual? If not, you need one as it will answer many of these basic questions. There is the Haynes manual, the BMW Factory manual (available free online if someone will post the link), and a host of other resources.
The cold start injector is #12 in the diagram below, and the wiring harness to unplug is #13
Best of luck sorting this out
Fred '74tii and '69 DCOE