The BL 1100 / 1300 were made at Longbridge, which is even closer. They had been superseded by the Allegro in '73, (some old stock wasn't registered until '74) but were the chariot of choice for the hardworking family guy on a tight budget. I don't know how much traffic you will need on BCB, but by their utter mundane-ness they would really suit the melancholy aura I think you are trying to create. Minix ones available on EBay.
Rovers and the booted variants of the Mini, less likely, they tended to haunt Sutton Coldfield or Penn, rather than BC. Singers, still many pre-Arrow series about (my mates mum had one). Imps were still in production, till '76.
Edited, having checked car production dates: 28xx