why not an airport with busy mixed traffic like Biggin or Gloucester?
I think the answer is obvious, is it not ?
Every man and his dog has probably at least heard of Heathrow, if not actually passed through it on business and/or holiday.
Only the Anoraks of the aviation world and direct users of Biggin or Gloucester would know about them and where they are. Remember also that at a later date the BBC may re-broadcast the programme internationally via BBC World (or whatever they call it these days).
The BBC needs to appeal to the widest possible audience, hence the choice of Heathrow and the dumbed down nature of the programme.