Well, you can do it, and it is not far - London is VERY SMALL compared to NYC - but trains here are not cheap, and cab prices are insane. Public transport in NYC is MUCH MORE AFFORDABLE.
And, after 1:00am, no more trains - only night busses. And they are often late, and full of drunks, and you may have to change several times depending on where you are going. I hate the night busses.
I go out to Chelsea, Fulham, Angel, and Shoreditch, all of the time - and I live close to Brixton, which is down south.
I go out all over London.
But, I get my booty on that last train... So, I have a transportation curfew, which is lame.