No study required the future prosperity of the UK is more important. Where - not Boris island as too far east and it would create major problems for the airspace design. It needs to be west or North west of London with high speed rail link and once built close Heathrow but keep Gatwick and use the Heathrow site for new houses and businesses
It's not just the airport that will need to relocate. Have a drive round the Heathrow area and you will see numerous businesses from hotels to logistics that all rely on the proximity of the airport. We are talking about tens of thousands of jobs here.
If you did close LHR you would wouldn't need to build houses or offices as the local workforce would have been decimated and there would be lots of commercial property available. You would however need to build them around the new airport location for all those who need to relocate.
The area for the third runway is actually mostly wasteground apart from the small area of housing of which BAA own quite a lot.