Heathrow and Gatwick are seperated by 25 miles, albeit 25 miles with bodies of water, areas of outstanding natural beauty, hills, roads, towns and so on
25 miles in 15 minutes is 100 miles per hour, average. The fastest speed a London Underground train goes is the Metropolitan line at 60 miles per hour. However, the Eurotunnel top speed is over 180 miles per hour, and it could be argued that that was more directly comparable.
Maybe 15 minutes is unrealistic, but it's not in the realms of science fiction. The question is paying for it.