Well, presumably AA doesn't reissue the ticket when a customer clears standby for an earlier flight. The ticket will still show first class, but the boarding pass will show coach. I think this situation is in that limbo between how AA does things and how BA does things, but OP is probably more likely to get actual data points from the BA forum.
My guess is that it will post as coach, but you may be able to convince BA to post it as business class. You probably don't even need to tell them the change was voluntary. But certainly I wouldn't expect business class miles/TPs and a refund of the fare difference.