Paxboy is correct, no new airport will be built in the south east in the forseeable.
Jabird's idea to expand LGW is probably the only alternative to expanding LHR although it is fraught with difficulty. It is reminiscent of the "second force" policy of the 1960s, 1970s and 1980s when it was government policy to build up LGW while criminally neglecting to sufficiently expand LHR, and putting us in the position we find ourselves in today.
Successive privately owned airlines were set up at LGW to take on state- owned BOAC and BEA, and despite being handed routes on a plate including those taken away from BOAC/BEA, were still unable to make a success of it. Everything was regulated and "bi-lateral" at the time so governments decided which carriers were on each route. British United, British Caledonian, Laker and Virgin atlantic were all intended as the "second force", LGW-based British carriers.
All failed except VS, and that was because VS saw the writing on the wall and got access to LHR as soon as it could. BD was always LHR-based and was a success for a long time, giving BA a run for it's money on shorthaul.
Even today LGW has relatively little transfer traffic, despite being a base for some very large airlines. It is an oddity and fairly unique:
(1) It serves as an overspill for long haul carriers who cannot gain access to Heathrow, e.g. Vietnam Airlines, and loses them once they do, e.g. Omanair;
(2) It is a base for point-to-point carriers such as U2, FR and BE;
(3) It is a base (secondary hub??) for BA for limited shorthaul and mostly leisure long haul (e.g. west Indies), again, predominantly point-to-point;
(4) It is a major charter/holiday company base (e.g. Monarch, Thomas Cook), again, point-to-point.
Like LHR it is almost full and has expansion issues, a new runway cannot be considered before 2019.
Not sure that non-Oneworld transfer traffic could be tempted to LGW. There is no incentive for the airlines to move especially as LHR is grouping the alliances together for ease of transfer.
It is possible that IAG may keep BD and BA separate (like IB and Vueling, or LH, LX, OS and SN, or AF and KL for example) because of integration, and staffing levels, pay, and seniority issues, at least in the short term, in which case there is no reason why BD would not remain in Star, while route swapping, codesharing and schedule synchconisation take place. IAG, Like LANTAM, could have airlines in, and make money from, membership of more than one alliance, especially if it buys more Star Alliance carriers, such as TAP.
New York is very different from London. Much of JFK and EWR traffic is domestic, as is all of LGA's, and transfer traffic is domestic-domestic or domestic-international. International-international transferring does not really exist in the USA, one has to arrive (visa or ESTA, border control/customs) and then depart (check in/security), which, of course, is to be avoided wherever possible. LHR on the other hand has 30-40% transfer traffic, much of it international-international.
Do like the idea of HS2 extending to LGW, would also suggest that it should also link to LHR. HS2 should also pass through London St Pancras rather than London-Euston to connect with the Eurostar, that would be joined up thinking. Still unconvinced that HS2 will be (or should be) built, but that could stray off-topic.