Overall, it's probably against BA's interests anyway to expand Heathrow runways. Doing so would only increase their competition, both EasyJet and Ryanair have discussed what they might run from there if only they could get the slots, and Norwegian would have a field day. Gatwick would empty almost overnight. Good for overall UK air connectivity and commerce, bad financially for BA.
It's certainly unreasonable to expect the current operators, principally BA, to make a significant advance financial contribution to the construction costs, if this only leads to a competition landslide and thus weakens their financial position.