The only two choices here are:
a. Build another two runways at LHR
b. Build the new Silver-Boris airport in the Thames Estuary.
Any airport can exist as a choice on paper, so:
c. Build a second or even third runway at Gatwick - not as commercially safe as LHR, but far less people to object. Over time LGW improves surface links and could even become the more dominant hub.
d. Build upto 3 new runways at Stansted, as previously proposed in 2003. It is a bit out of the way, then again so is anything in the Thames, so the construction risk is far smaller.
e. Do nothing. Not as stupid as it sounds, considering that airports are give and take, and given the likely challenges the industry faces (environmental, cost of fuel, tax gouging).
Remember that however much you dislike c. and d, they are still better options than fantasy island because any capacity added there (and of course at LHR) is a
direct add and on
dry land.
Faced with building on water, AND having to create 2/3rds of the infrastructure just to replace what will be lost by closing LHR, it is easy to see why the island is a non starter.