As I understand it, BA originally wanted everything out of T4 and wanted it to be theirs alone but this was not allowed. Further, there were too many flights for the facility.
Now they want everything out of T5 but there will be a few operating from T3, to facilitate One World code shares. The 'too many flights to fit into one terminal' problem was fixed by reducing the number of routes.
Largely, however, they will be masters of their own domain and are clearly expecting great benefits. As to why other carriers have not complained and done more to get BAA to change? My guess is:
1) Corporate complacency.
2) There was no public outcry. Pax used EGLL because they had no choice.
This was changed by:
1) 9/11 and the 'security' regs that pushed the place well beyond the limit and the multitude of smaller problems that had existed for years, got pulled into the lime light along with the new problems.
2) The LCCs made domestic and continental flights cheaper
and obviated the need to use LHR. At first, the carriers saw a reduction in short haul capacity and cheerfully used it for new services or long haul. Rather too late, they found that the problem was bigger than they thought.
In corporate (and domestic) life, big problems usually start about 20 years before they are big. The roots of the problems are clearly visible but will take many years to clear. Mostly, they will be cleared by people not using the place.