The answer is not to expand Heathrow, it's to expand West Midlands or Birmingham and run more trains to/from London.
People are already staying in towns like Rugby and Tamworth when they come for business in London, or moving there, because the commute is less than from a Hotel within the city.
So why not land the planes an hour (by train) away, you'll hardly notice the difference between that and the time you spend standing in queue with your shoes in your hands.