In reply to j3nesis:
Professionals will be able to give better advice than I can, but the way I think about it:
* Climbing involves pushing with legs (which is quite easy) and pulling with arms (which is hard)
[ although good technique minimises the amount of arm-pulling you have to do ]
* Therefore the most important muscle group to "strengthen up" are the arm pulling muscles
[ finger strength is also important, but a very easy muscle to injure, and realistically get stronger just be climbing ]
* There were some articles on here recently by a physiotherapist saying that it's important to train the antagonistic "pushing" muscles too, as you can injure yourself by having one half of an antagonistic pair a lot stronger than the other
As a side note, if the gym is set up with a line of auto-belays, do be careful! Nothing wrong with auto-belays, but if you're focused on a gym routine and don't have a partner, it's easy to forget to clip in (which I think is the most common cause of auto-belay accidents). With traditional "partner checks", if you do something stupid, it's normally picked up by the other brain standing next to you.
There's also loads of other places to climb in London, and loads of people who can give you lifts to the real world