I wonder if perhaps that this is a problem that has no solution? As a Yank I hedge a bit of admiration for those who are tackling this issue but do the geographical realities of greater London simply prevent a magic bullet solution? One simply might not be able to find enough land (or Estuary) space onto which to plop 4 runways/terminals/access roads & rail etc.
Maybe a stretch with the analogy, but New York and Washington have the same problem. Domestic passangers prefer Laguardia but the international has to be handled out at JFK which, as many readers here know, adds a LOT of transit time to downtown Manhattan.
Washington has Dulles well to the west of downtown and domestic passangers doing business in the city prefer National.
Short of building a runway 18 at LHR through the resevoir what, at this point, can be done?
If I were King for a day I would tax the masses and build an elevated motorway from LHR to downtown. I hate for my memory of life to be sitting in traffic staring at the sign for the Hammersmith flyover...
That all having been said, I would like to see a Boris island solution. Clear out Westminster (no need for the MPs anyway) and pour the runways there.