I'm struggling with the fascination that following the take over that BM & BA would be integrated.
IAG will at that stage own 3 airlines so why must they integrate two of them? If that were the case there would also be a case to integrate BA and Iberia.
In some cases it could be seen as a better option to keep the 3 airlines as seperate operational and financial entities to have 3 potential profit streams.
If nothing else I doubt BA would have the resources to fill the LHR slots with aircraft and crew and moving staff from LGW would be a logistical challenge and costly.