Quote from public press release by Government.
Expansion costs will be paid for by the private sector, not by the taxpayer. It will be for the Civil Aviation Authority (CAA), as the independent industry regulator, to work with Heathrow Airport Ltd and airlines operating at the airport, on the detailed design and costs to ensure the scheme remains affordable. The government expects the industry to work together to drive down costs to benefit passengers. The aim should be to deliver a plan for expansion that keeps landing charges close to current levels.
So no cost to UK government - apart from the cost to move off the balance sheet to debt and no tax paid, keeping the dividend intact.
