As AA is the ticketing carrier, you must deal with AA until the day of travel, at which time BA can deal with the issue.

As CWS notes, AA can request space from BA, but it is a liaison and not automated process. Thus, this close in not really worth it.

For what it is worth, APD is based on your class of service departing from the UK. While it seems unlikely that you paid any APD on a conenction (<24 hours at LHR), it would not matter unelss you were booted out of the premium cabins on your AA flight departing LHR.

As soon as you are within 24 hours of your originally scheduled departure, call BA and ask to be moved to the next day's service. Have that flight number handy so that you are making a clearcut request. Given that other CW passengers may have been moved to the next day, it may possibly be full. I would do the research on other flights and have those handy too.