You might have some a leaky or starting to fail fuel injector. On 0ne of the forums I checked, someone said BMW's run rich at startup, then lean out to a normal mixture after the engine starts up. So if you have one marginal injector, it could be dumping too much fuel into the cylinder, making it hard to start when cold.
After the engine warms up, the engine computer reads the temperature and will not need to deliver a rich mixture. Get a bottle of Chevron Techron fuel injector cleaner at an auto parts store. Next time you need a fill-up, pour the bottle into the near empty fuel tank, then fill the tank with gas.
Techron is a great fuel system cleaner and it will do a good job to remove any fuel deposits or varnish inside the bad fuel injector. See if that does the trick. You might have to repeat a second time.
You can also take your car to shop that can do a pressurized fuel system cleaning - they will inject a high strength solvent directly into the fuel injector rail and basically do the same thing as the Techron, but at a higher pressure. You also should take your car to a shop that can download ('pull') any fault codes from the engine computer. There might be one or more fault codes stored in the memory that will give more clues if another item needs to be checked.
Sounds like fuel filter to me. When was the car serviced last? Try a bottle of injector cleaner, you may have build up on throttle body also.
There are various products which claim to clean throttle bodies but the best way is to manually clean it usually a mechanics job.
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