The figure of £6 billion public expenditure for R3 needs to be put into context. It relates to the estimated cost of surface access, including roads and rail. Davies has stated that the allocation of this between the scheme promoters and the public sector would be a matter for negotiation. For some reason, however, his 'base case' numbers assume that it is funded by the public sector ( ie the taxpayer).
In the supporting documents on the website alternative scenarios are considered. One of these assumes that the scheme promoters meet the full cost of the £6bn surface access works. It calculates the impact on Heathrow's airport charges, showing that the swing of £6bn from public to private only adds c10% to HAL's projected airport charges.
I think it would be safe to assume that HAL, as the scheme promoter, will end up having to swallow these £6bn costs. That would be my view anyway. Recent precedent is that HAL has had to foot the bill for surface access costs, at least in relation to extending HEX and the Piccadilly Line to T5. I can't see why this principle shouldn't also apply in the case of R3-related surface access costs.