Some thoughts
Heathrow is a hub airport, that means airlines can use it as an interchange between long and short haul but also between transcontinental and transoceanic. They will move (and are moving) to other hubs if the capacity is not available. Telling passengers that they are going to fly into Heathrow then get on an hour long train journey to Gatwick/Stanstead/Luton or a 3 hour journey to Birmingham or Manchester will not work. You would not fly into JFK to pick up a connection at PHL. Nor will any SLF.
A European hub is a business operation, Amsterdam, Paris, Frankfurt, all want the traffic that is currently being refused slots at Heathrow. Heathrow will die if new capacity is not added. Leaving BA with its own private airport and the rest of the UK airports being regional feeders to the favoured hub in Europe. This is happening already ably assisted by the Governments Air Passenger Duty.
UK by geography is in a good position for transatlantic flights to interchange but with the longer range of the newer widebodies that is ceasing to be such an advantage.
A new airport can be built from scratch - i.e. from first design to operation in 10 years (as was the case with Hong Kong). An airport outside the 12 mile limit but inside the Thames Estuary with road/rail tunnel access to both North and South banks could have as many 'Mulberry harbour' type runways of whatever length as it wanted. No noise abatement problems, no continual overflights of central London, no planning issues outside the 12 mile limit apart from the North and South bank terminals. The South terminal linked into the TGV network. There could even be interchange with merchant shipping and river boats to central London. There is lots of money looking for a reliable long term investment more than enough for a private venture to be able to achieve full funding. The new SESAR/EUROCONTROL Step 2 plan envisages no air routes from TMA exit to TMA entry so the current mass of air routes will cease to exist around the time that the airport would go operational if started next year.
This should not be a government decision its a business and currently BA and Ferrovial are controlling the government for pure business reasons. BA (IAG) do not want any new slots at Heathrow as that could lead to more competition. I expect the can to be kicked down the road repeatedly on Heathrow. Then when a decision is made the planning battles will continue for decades.