As Andy S has implied, trying to build a new large airport without closing Heathrow just wouldn't work. However if it was closed, AND the necessary infrastructure such as road and rail was in place when the new airport opened, instead of a few years later then it could work. WHBM mentioned the lanes heading west on the M4 out of Heathrow - bear in mind that a lot of that traffic then swings north for the M40 and M1, so an airport to the east of London that had efficient road and rail links serving the north would be viable. The traffic isn't all heading for Wales and Cornwall!
One thing we all agree on - it'll never happen for both political and cost reasons.