Either build a third runway at Heathrow or look at relocating it to the north west of London in Beds/Bucks as was seriously considered back in the 1960s. You could link it to the mainline from Euston or Marylebone.
I found this reply genie lay interesting but puzzled why it would practical to link it to the chiltern line when (from memory) the Oxford line runs very close to heyford.
The Chiltern line lies closest to Heyford as you used to see the landing lights from the train (I mean, of course, the late lamented Paddington to Wolverhampton via High Wycombe). It's very close to Ardley tunnel
The Upper Heyford is in the triangle of the Chiltern and Oxford lines just South of Aynho Junction. The village is nearer the Oxford line which has a Station. The East end of the runway practically adjoins the Chiltern Line and the West end is about a mile from the Oxford line.