Radical solution
You only have to overfly LHR to get some appreciation of the difficulty of expansion. A very radical solution would be to start afresh at a site (now there's the next prolem!) which would permit a proper way to provide the airport that London actually needs. It would cost a phenomenal amount and would no doubt be a planning nightmare but it could be done. With a fresh start the infrastructure needed to make it work could be planned for including fast rail link to London as well as the surrounding region, enough runways to cope with forseable (always short sighted so far) demand plus an appreciation that more room may well be needed in the future. The existing site at Heathrow must be worth a fortune in terms of redeveloping and so goes some way to aiding the cost and whatever was built there would be better for the residents. So far only sticking plaster solutions are put forward whch are not going to work for long. Admitedly if you take all the surrounding "London" airports into account there are a lot of runways in the larger London area but where do people from outside Europe want to fly to when they think of London? Of course they are only given the choice by the airlines.
It's not just London of course. We seem to have been reluctant to bite the bullet in this country when it comes to providing the infrastructure at regional airports. Compare BHX and MAN with the likes of MUC, FRA or HAM. At MAN the solution was to build a second runway that can only be reached after queing up to dash across the first runway and then can only be used one way for take-off and the other for landing.