My experience is over a decade old so things may have changed. Also, my experience was in moving from the UK to the USA so I can't speak knowledgeably about banks in BDA.

The one piece of good news is that you should not lose any Avios. The 20K+ that you have accrued should be sitting with the bank in Bermuda and once the problem/mismatch is resolved all the previously accrued Avios should flow into your account. As well as the address, it's essential of course that the BAEC number held by the bank in BDA is correct. It's worth double-checking that but I doubt that's the problem.

Is your UK credit card a BA Amex one? If so, if your BAEC address is now in Bermuda you are correct that those UK BA Amex points should not be transferring. If they are still transferring monthly it would suggest the address has not been updated in the BAEC system. When you speak with BA again you might want to tell them that your UK points are still being transferred which is a large clue that the address change has not registered successfully.