Many people do the LHR-LGW (and reverse) connection every day without issue and catch their connecting flights this includes clearing UK immegration and collecting their bags.

If BA had an issue with lots of passengers missing flights then they would increase the MCT.

Whoever told you that UK passport control can take "several hours" is talking absolute nonsense.

Your flight from Berlin would have to be significantly delated (a lot more than 15 minutes) for BA to even think about rebooking you automatically from LHR.