I just don't see using the WLL is very likely, given the capacity issues either side of the connection (well, on it too). A Gatwick to Heathrow connection will definitely be needed (it is now) looking at how the Davies Commission is shaping up. It must be remembered that airport to airport traffic is not likely to be large, but incorporating such a connection in a wider route to provide better connections to the airports is more likely to have a good business case, i.e. the trains would be well-filled.
I suppose transfer pax would need to go through immigration checks at either end but I think they do already, in the sense that airline staff just don't allow anyone on flights!