Britain already operates some of the most heavily used, densest and most efficient transport infrastructure in the world. It cannot "efficiency increase" its way out of this conundrum unfortunately, and that means building something.
I would say that a Thames Estuary Airport would be a good option, however, this would I think bring it into conflict with Schiphol Airport and for that reason alone may not be feasible.
Building a new International Hub with three runways and automatic expansion to add a fourth, well, on environmental grounds alone that will never happen. This leaves finding a way to operate Heathrow and Gatwick essentially as a single Airport, however, given that Gatwick is ALSO operating nearly at capacity, there are limits there, even after the existing Northern Runway is shifted a fee metres north to enable simultaneous operation of both runways.
No matter what decision is made (and as someone who works in Aerospace, I favour rail expansion) we are going to piss somebody off!