This could be technically more difficult than it seems because Terminal 5 was set up exclusively using British Airways booking systems and it was a devil of a job a few years ago to get Iberia in to work as despite having the same ultimate ownership they had different legacy systems. Maybe it kind of works through the code share or things have moved on (on a search looks like they have needed to bring in extra kit so AA can only operate from a limited part of the terminal for check in etc).
If BA was running anything close to a pre-March schedule they wouldn't fit in 5, let alone anyone else. Then again AA is running a much reduced service as well. However, since AA and BA effectively operate and plan the transatlantic services as if they were a single airline it has been an aspiration to get AA into 5, even if more BA services went back to 3. So people may end up doing the shuffle in the reverse direction.
T3 may not come back into use until the winter schedules at the earliest though, and T4 is not expected to reopen this year.
EDIT: Finnair are also moving to 5 later this month.