Don't worry about "fault" as that is irrelevant. You check a bag in and you get it back. If you do not, it becomes the responsibility of the last delivering carrier to either locate and return it to you or to compensate for its loss up to certain limits.

If, when you arrive at LHR, your bag is not there, you will file a report with BA and BA will make arrangements to deliver the bag to you at your hotel and will issue you a claim number which you can look up in World Tracer.

Do not spend an instant thinking about how it all works, how you booked the flight or how AA or BA choose to make all of this happen. It is designed so that you need not.