They don't serve that route, but they do fly from three airports in London (Gatwick, Stansted and Luton) and from Brussels.
I have wasted too much of my time in delayed trains trying to get into London to believe that.
And based on travel times you could add some other airports to that list of London Airports. Southampton Airport is faster to get to than Stansted or Luton for a lot of Londoners. And with HS2 BHX could probably change its name too.