Its not just about moving Heathrow, there are many businesses around the Thames valley that are there to cater for the airport. The people that work at Heathrow and these other businesses mostly live in and around that area. Closing Heathrow and building this fantasy island would mean moving the whole lot to the other side of London.
Crossrail will take people from Canary Wharf to Heathrow in 30 minutes. There is actually quite a lot of wasteland around Heathrow if you have a look. The third runway proposal seems to me the way to go. Unfortunately this would mean bulldozing a very small village and moving a few people. It is however a much more feasable plan than relocating the huge numbers as above.
I cant help feeling that the whole thames island plan is being proposed in order to show just how ludicrously expensive it will be so that the government can then revert back to the third runway plan and act as though it is saving money.