There are two relevant numbers.
One is your NHS (National Health Service) number. This you need when you see a doctor. If you can remember who your last doctor was, call and they will be able to tell you this. Otherwise, contact the local health authority of where you last lived. It would help to have this when you register with a doctor but it is not an essential to receive treatment.
The other is your National Insurance number. This is the number you need when you apply for or start work. You can get this again by calling HMRC (Her Majesty's Revenue Service, formerly the Inland Revenue). Once you get here, you call them, they will send it to you.
Vicky