I had a problem with AA not long ago LHR-ORD, unfortunately the damage didn't manifest until I undid my suitcase and the lid fell off.

I called AA who said 'Bring it to the airport and we'll see'.

I had 2 days in ORD visiting my elderly father, so I made an executive decision not to bother and buy a cheap suitcase locally...I had the feeling that I would be travelling with my belongings in a box by the time it was sorted.

My understanding with AA is that if you have left the airport before reporting the damage it will be in 'the lap of the gods' depending on who deals with you, hopefully BA will be better as reported up thread ....good luck