I used to babysit from when I was 14 to when I was 18. I started off charging £4.50 or £5 an hour and that went up after a while to £6 an hour. I rarely babysat after midnight so I don't think I had a special rate for that. I generally left it up to the parents if they wanted to give me more. But then again I never had a driving licence although I'd had some first aid training. The area I live in sounds much like yours, OP - not London but still a relatively affluent area. I wouldn't charge more than £7 per hour to begin with, especially if you have not babysat before. I only felt able to charge more after a time because I had become more experienced with the kids.
As for what I used to do - occasionally I would have to cook dinner for the kids but that was really rare. Mostly it just involved putting them to bed, making sure they brushed their teeth etc, reading them stories. Older kids (10+) tend to put themselves to bed quite nicely though. The really tricky part is dealing with bad behaviour - getting them to go to sleep, getting them to go to bed generally, how to cope if they're missing their mum and dad or usually have a different babysitter. Sometimes kids are just generally naughty as well/will mess about so you need to have tactics up your sleeve to deal with it (distraction techniques, making going to bed a pleasant experience etc).