I have a bmw 328i 1996.when you start the car in the morning when its cold the engine accellerates up an down for a few seconds

Call out a local mechanic as without my head in the engine i couldnt tell you anything ,might just be the fan belt broke but ?? from afar i cannot know

This may be the problem of the anti-theft system is on since the security doesn't recognize your key.Your key may be faulty or your alarm is armed and the vehicle can't start.

check your fuel pressure, good luck

I do not know of an "Air Pump" for a 1996 328i. Did you mean
fuel pump (15A fuse # 18) or
blower motor (30A fuse # 20)

is it turning over.
could be a faulty main fuse.
problem is you cant see in the fuse to see if broken.
when cold points contract making it start, when warm it expands and creates a void

I would change the oil, disconnect the coil or fuel pump and crank the engine for several minutes to let the oil pump work.
Put a few gallons of fresh gas in the tank, and see what happens.
You may have to run that gas out and then replace the fuel filter.
If it starts ok, just let it idle for several minutes before you raise RPM. Listen for noises and see if the cooling system works.
There is a chance you may have to change the trans fluid and filter after you try to drive it.
And after driving a few miles, be sure and check the brakes and all of the engine hoses for leaks.

take it to a parts store they will scan it for free the code can lead you to the problem

possible air mass meter fault or split concertina air hose between airr flow meter and manifold

Every car needs a heating engine time specially when its cold, I recomend like about 5 to 10 minutes after you start engine and before you change from P to D. If you try to start the engine first time in the morning, it's cold outside and you immediately go on run you are going to have this trouble every time. Every engine has it's operational standard temperature a 5 to 10 minutes working engine, car stopped, may guaranty this standard temperature and optimal function conditions, and also keeps a longer life of your car.

Good luck

water pump may be the way to test take off cap when cald hold reves see if water flys out if so ok if not water pump maybe on the way out