We're not going to get a 24-hour Tube in London in the near future.
The New York Subway has multiple tracks on several lines, allowing overnight maintenance to take place on one while trains run on the other. The lines are also cut-and-cover style, meaning that they're close to the surface and fairly roomy.
In London, this is not the case. Most lines are small, deep tunnels in the centre, and there are very few points where there is more than one line in each direction. Overnight maintenance - from cleaning up litter to track inspection - cannot currently be automated, and the nature of the underground power supply means that it's not particularly easy to isolate small sections of the tunnels to allow trains to run on other sections on a vastly reduced service.
Perhaps one day...
In the meantime, the night bus service in London is excellent. I used to have a choice of over 10 buses an hour, all through the night, when I lived in Stockwell, and used it all the time. People can be oddly snobbish about buses...