Thanks for the quick replies so far. To clarify, I'm not asking any airline to issue a BP to someone without a passport. I'm only wondering if anyone has actual experience with this kind of situation (travel documents do not arrive in time). It would seem to be similar to the "Flat Tire Rule" on domestic flights. However, this being international, I don't know. That's why I'm asking if anyone has relevant experience.

FWIW, the courier service had no problem scheduling a delivery to occur at LHR. They claim they deliver last-minute documents to airports all the time. My main concern is what does AA do if I'm not ready to catch my flight, but I can catch a later flight that same day.