It's a very bad outcome from you, but most unfortunately, it is the risk you take with 2 separate tickets.

It's not ultimately a BA issue, in that while they issued the tickets, the on-the-day airlines are AB and AA. BA's ticket required you to be in LHR by T3 and it wasn't really BA's fault that you didn't make it. AB will say "you paid us to get to London, and we got you there, eventually". AA - normally - are accommodating in these circumstances, and would rebook, so I am not quite sure what happened there.

If you split tickets like this, you lose protection, that's quite well known and certainly often discussed in this forum. You have to mitigate it by insurance or self insurance. Or you pay more to get a through ticket, that extra cost is your insurance. I certainly don't envy your position and I would have hoped AA could have been more flexible, there is supposedly an AA policy to protect oneworld bookings, which used to be public but now is not. May be a silly question, but is the AA agent aware that AB is oneworld? At this stage of the game they own your ticket and they hold the cards.

[Calling BA was never going to work, the telephone agents don't have any flexibility in this situation, this is a classic example of something that can only be sorted out at the airport].

One specific thing you need to do is to persuade AA to protect the return sector, which they should be able to do.