London is as safe as a big city can be. I live outside London and everytime I go for a nite out in the city I end up waiting around Victoria station to catch the train home at 3 or 4 in the morning (normally by myself), and never had a problem. Yes you will see dodgy people, but you see them 24h a day! Normally it is really safe around there (i think the bus for stansted departs from victoria coach station which is next to the train one), and if i was you i rather be around London until 3am than sleeping in the floor in stansted (but that is my personal choice!)
Anything else, let me know!
x x Miromar x x