Presuming that you were indeed ticketed LHR-JFK-SFO with the second segment operated by AA, BA was indeed correct to require you to make your report and deal with AA. As the last delivering carrier, that has been industry practice for 50+ years and for better or worse, it works.

As others note, a bag is not treated as lost until the 21st day and the vast majority of bags turn up in 24-72 hours (depending on destination). There are, of course some bags which have been stolen or destroyed and therefore will not be found. But, those are the exception.

In the meantime, the Montreal Convention covers your wife to fair market value of approx. EUR 1,100. AA will reimburse for reasonable expenses up to that amount and, if the bag is ultimately lost, the reimbursement will reduce the total pay out. Your travel insurance may be more generous and may simply provide a flat payment which gives you certainty.

In the meantime, when you reported to AA, you should have been issued a WorldTracer number. You can track that number online and you will see the same information as AA sees. E.g., when the bag is located and its progress to you.

Not to berate OP, but for the benefit of others, given generous carry-on allowances, it is always a good idea to have the first 3-4 days of clothing in hand. This takes all of the stress out of the situation and also provides plenty of lead time if indeed the bag really has gone its own way.