Originally Posted by Often1

OP - what happens depends on the situation:

1. If BA arrives on schedule, it is AA which would need to rebook you. AA will likely do so as a matter of discretion, but is unlikely to rebook you onto other carriers and even less likely to pick up the cost of your hotel and food.

2. If BA causes the misconnect, it is BA which must rebook you. EC 261/2004 will apply and will provide a hotel room and food if necessary. Depending on the reason for the delay, BA may also owe you delay/cancellation compensation. BA may also be willing to rebook you onto UA.

3. However, do not presume that there will be availability on the next AA or UA flight. It's the middle of the summer and flights are often overbooked. If nothing else, the decent seats will likely be gone. Thus, the service recovery is not likely to be particularly pleasant.