I quite like this idea.
Not that it is very feasible, but one reason why I like the idea of a very fast rail connection between Gatwick and Heathrow is that I was saying ten years ago that we should have one. During the last government’s study of runway capacity and expansion there was a lot of talk about hub airports and how Heathrow with its number of terminals and runways could not compete with Schipol, Roissy and Frankfurt, but a shuttle that takes less than 10 minutes to get between airports effectively makes Gatwick and Heathrow a single airport. It took me longer than that to get from one gate at Singapore airport to one at the other end once.
It was always unfeasible, since it would need to be a deep tunneling operation like the channel tunnel and even then would be under a lot of the richest areas of the UK, also the greatest benefit would be if it was a free or very cheap service – who would be choosing London airport over Paris if there was an additional £20 to get from one terminal to another?
Such an idea would be a lot harder to do now than it was ten years ago when Heathrow and Gatwick were owned by the same company.