Easy, you build a new London Gateway airport in the Thames estuary
Re; avoiding the need to overfly built-up areas, won't the Thames Estuary airport have runways aligned East-West like LHR? If yes, they'll still be a fair degree of over-flying London of approaching from the west. If taking off eastwards, the North Atlantic traffic will still need to turn west and overfly Greater London.
Correct me if I'm wrong but I don't see a vast difference in overflying London for either LHR or the Thames estuary airport.