IAG would not be necessarly all that bad a fit.
IAG has a presence in both the Regional and LCC markets.
BD Mainline - obviously this would be integrated overnight into BA, I think the majority of the mid-hauls, along with BHD and DUB would be kept, short term at least. EDI, ABZ and MAN would be the first to give their slots to new routes, Its also hard to say how many, if any of the slots LH currently uses to feed hubs would return to BD. In any case these may count as 'divested' slots.
I could possibly see all Domestic/Ireland service moving back to T1 to make room at T5 for the mid hauls and other new services.
Baby - IAG has a low cost airline in the shape of Vueling. Many of Baby's biggest markets are in Spain, AMS, etc which are Vueling markets too. It would make sense that these two airlines work together, especially in a situation where the "bmi" brand is disappearing. Go Mk 2 anyone?
Regional - this is slightly trickier, but there is the possibility that Regional could work with BA Cityflyer or be sold off separately to flyBe.