This can be done at check in, or if not going via check in at the lounges, or there is a service desk in T7 near gate 6, or failing all of that at the gate. Usually those with outstanding document checks are called to the service desk before boarding. The lack of a document check should not prevent the bp being issued.

The LA flight arrives around 0930 - the OP has got quiet a wait before the BA flight as the first departure to LHR isn't until 1830!