without yet getting into which airline was responsible, one thing to know is whether your friend had one ticket number that included both the AA and BA segments, or two separate ticket numbers i.e. one for JFK-BOS and another for BOS-LHR.

If the former (1 ticket for both segments), then the bags should definitely have been checked through to LHR, and the bag tags would reflect that final destination. There should be no question if only 1 ticket number covered the whole journey that AA check in somehow messed up her check in. She should write to AA customer service concisely explaining the issue and requesting a check for the ticket value, if not additional compensation for the inconveniences.

If the latter (2 ticket numbers per person), then she (and similarly her daughter) were responsible for checking in with their bags at BOS at or before the closing of the check in window. However, since AA and BA are oneworld partners, in such a situation, they are often able to check through bags, check you in on both flights anyway and provide boarding passes as a courtesy. In this case, it seems to me that the bag tags provide strong evidence of whether or not your friend was actually told that she and her daughter were checked through to LHR. If the bag tags only show BOS as the destination, it is probably going to be tough to convince AA or BA that they were at fault for your friend not providing enough connecting time to pick up bags at BOS and recheck in for LHR. They are likely to claim that the AA check in meant that they needed to check in at BOS (not just pick up boading passes) and that your friend misunderstood. Even if your friend heard correctly, it's a he said she said situation if the bag tags only show BOS as a destination. Your friend has no other proof than those bag tags if there were separate ticket numbers for each flight. Hope she kept them!