I've always thought that if Heathrow is to have a third runway then its has to be at Northolt. You would need to realign the existing runway so that it is parrallel with the the two at Heathrow. This could be done within the existing perimeter.
You could then build a terminal over the existing Central Line depot which could be replaced by some underground sidings - as has already happened at White City. This would then be opposite the Central Line (which could be extended to Uxbridge) & Chiltern Lines. At 1850 m it would be as long as DCA (Washington Reagan National) & you could reach most of Europe.
Problems
Noise & pollution - might need to wait for the next generation of aircraft.
ATC congestion - no worse than a third runway at LHR.
Getting between terminals - not much worse than LHR T6 if an underground railway is built. The larger the airport the longer the taxi (JFK comes to mind but there are others). With a single runway & terminal taxi times should be shorter. There is no reason why a transit system cannot cope with international & domestic - Frankfurt & Seattle have separate vehicles for the two in transit trains.
More ground congestion. However you would get a terminal adjacent to two rail lines. You would need to encourage some users of the A40 to switch to rail as it is not feasible to widen the road. No worse than LHR Runway 3 / T6 though.
Advantages
You effectively get an extra runway & terminal without having to acquire any extra land other than from the MoD & LUL. Just imagine the protesters having a field day if some old dears refuse to vacate homes acquired through compulsory purchase.
Verdict
Currently complaints about noise from locals would make it a non starter. Boris Island with LHR remaining open would appear to be the best option. £10bn to build an airport is affordable, £50 bn including all kinds of infrastructure, four track high speed lines, etc, would be hard to justify. It might make sense if you could use existing spare capacity on HS1 (and there is plenty).
However as other posts have said it might only work if LHR were closed. What would that do for the economy of west London. It might give local MP John McDonnel and the London Borough of Hillingdon pause for thought. If airport workers have to move and support workers are likely to move their jobs people will start getting worried. (You could start on threat on whether you could have a two hub airport strategy for London. New York manages it but thats about the only example I can think of.)
In that scenario a backdoor third runway for Heathrow with a strict night curfew and limited to quiet aircraft might be preferable to moving LHR to Boris Island.
You will remember that LCY was limited to propjets when it opened and was on no account to accommodate jets. We know what happened. Similarly start with domestic only. Then add Ireland. Then CDG / BRU / AMS and go from there.
Perhaps its a shame that BEA moved to LHR in the 50s. If a railway tunnel between the two was built we would have a third runway at the moment!