If BA is the operating carrier then BA is who you should be talking to. However, given that AA and BA will both have your flight in a different PNR it is highly likely that both will rebook you, and probably on different flights (as in the story above). So, in fact, you should probably talk to both as (in my experience) there is a real chance of return flights getting cancelled when you don't take one of the rebooked flights.

In my case, I missed a LHR-PSA flight on an AA codeshare and AA booked me on the next PSA flight (24 hours later) whereas BA booked me a few hours later on a BA flight to FRA and a LH flight to FLR. I took. the BA rebooking and then a few days later discovered my whole return trip had been cancelled by AA because I hadn't taken the PSA flight. Fortunately they were able to reinstate it all (except my seat assignments).