Birmingham European merged with Brymon to become Brymon European around 1993, this was then demerged soon afterwards with the former Birmingham European becoming a BA franchisee yet owned by Maersk, trading as Maersk UK. This was the norm until around 2003-4 when the franchise was not renewed and the company was relaunched as Duo which then went bust.
Anything else at BHX with BA was British Airways (Regional) which was really just BA.