It won't be Government (ie our) money. If BA aren't prepared to pay then I'm sure other airlines will, given the yields available at LHR. It's all a bit of a bluff by BA, who (as has been noted) are terrified of easyJet.
But I do agree with BA's argument that the project is unnecessarily expensive. It's a rerun of T5, with the regulator (CAA) failing to get a grip before the costs get out of hand.