I don't think bmi will keep BMED as a separate entity for too long, they already have three separate companies to look after (not counting cargo or charter) so adding a fourth would be too much. I'd expect to see BMED incorporated into bmi at the earliest opportunity, with the BA franchise ending also as soon as contractually possible, though I wouldn't be surprised if BA wanted to codeshare on these routes.
I can't help but agree that bmi would be well placed, as they move into the longhaul foray, to once again have a serious look at their brand, including re-establishing the kind of British style they put into the company when the trans-Atlantics were first launched.