Farnborough airport is a far better idea. Ready to go as is
Farnboroughs runway is 8005' long - which is only just long enough to get a 737-800 fully loaded with bucket and spaders off to Benidorm. A fully loaded 747 or A380 needs 10000+, and preferably 12000 ish. There are plenty of 7500-8000' runways in the London area that could be used for domestic and short haul, but these are all useless for long haul. The only 10000+ runways currently available in the south east are Heathrow's 27L/R (both 12000+), Gatwick's 26L (10879') and Stansted's 22 (10003').
Gatwick does, notionally at least, already have 2 runways as the parallel taxiway can be used if required. But 26R is only 8400' long (useless for long haul), and is only 200m from the main runway, so cannot be used simultaneously. Minimum separation for runways in parallel use is 1300m. Heathrow's two current runways are 1400m apart.
IMV, both Heathrow and Gatwick should get another 10000' runway. The only other option is to adopt the Munich/Berlin/CdG approach and build a new super airport somewhere to the west/north west of London in green belt land. Boris Island is in the wrong place - impossible to get to from most of the UK.
Not going to happen any time soon due to NIMBYs.