Not everything is connected by HSR. Getting from Heathrow to city centre can take ~1.5h on public transport, and that's like ~15 miles in straight line.
If you're building an airport you can presumably also build other infrastructure. I am not saying it needs to be reach it in 20 minutes via HSR, just that distance is not the problem infrastructure is.
Dulles airport was built 30 miles from DC in what was the middle of nowhere at the time even though DC already had an airport. "In 1965 Dulles averaged 89 airline operations a day while National Airport (now Reagan) averaged 600 despite not allowing jets." Dulles still does not have a subway link to DC, and it's not the closest airport yet it still sees 21 million passengers a year.
London like most major cities already has an airport. However, expanding airports means addicting a runway ~1.2km from existing runways which causes more noise issues as people are now effectively 1.2 km closer to the airport.
So, sure expanding existing airports is an option to consider, but doing so has a real and measurable negative impact on property values.
PS: I am not trying to promote a new airport further from London just saying such a thing is an option for London and most other major cities.
Dulles airport was built 30 miles from DC in what was the middle of nowhere at the time even though DC already had an airport. "In 1965 Dulles averaged 89 airline operations a day while National Airport (now Reagan) averaged 600 despite not allowing jets." Dulles still does not have a subway link to DC, and it's not the closest airport yet it still sees 21 million passengers a year.