the airline that had 727s on the American register but appeared with one example sporting British Airways titles
Aha. That was American Trans Air in summer 1988 with an old United 727, It was used mainly on the BA Shuttle from London to Glasgow as a Backup aircraft when the first section was full. It was leased at a huge hourly rate, but the backups normally only did one or two trips per day. BA regularly seemed to go through aircraft shortages at this time, and it was to cover these (incidentally, the last Trident went at the end of 1985).
It also featured back at post #2957 back in this thread, and was immediately answered, even with a photo.