BA need the slots at LHR to continue to compete with the other large European carriers and Emirates, all of whom have much more room for expansion at their own hubs. I would be very surprised if BA did not acquire the majority of the slots. No doubt, some of those slots would have to be shared out to others for the purposes of competition regulations but I am sure BA would be happy enough with that.
If slots must be used straight away, then a total takeover of BMI seems logical as the aircraft and crew are there to continue flying whilst BA sets about a restructure of its routes and timings.
There is, of course the possibility of benefits being seen across the UK. Operations into the likes of MAN and EDI could be streamlined, potentially with larger aircraft releasing further slots at Heathrow that could possibly see the introduction/reintroduction of connections to other UK POD's. Thus, British Airways could start to truly think of itself as being British again.
With regards to Star Alliance, though they would lose a chunk of Heathrow, it would not mean that they no longer have a route in and out - plenty still serve it - it would more likely have an effect on transit with the emphasis more on Star strongholds such as Frankfurt, Brussels and Zurich.