The Brighton to London trip is just over 50 minutes on the fastest currently available route, which has two stops (East Croydon and Clapham Junction). The BBC article linked in another comment [1] mentions that the train used to make this video was a specially commissioned non-stop trip, so I expect it took a bit less than 50 minutes (assuming it was going at the same speed, of course).
[1] http://www.bbc.co.uk/news/uk-england-23853863