Llyods give them out to people at the age of 11. There was a big fuss over it last year. Nationwide don't until you are 16. Natwest give you one as soon as you open an account that allows one, they give Solo's to under 18's.
Credit cards are 18, if you go to uni and open a student account the banks will throw them at you. I never took one on principle, I never want a credit card.
Credit cards are also subject to credit checks, if you have a poor credit rating you won't be able to get a credit card.
You can get your own debit cards from certain banks from the age of 10, but your parents will still need to be on the account (albeit without access).
For a credit card, you can get an in-house credit card from any bank from 18 - for the platinum cards and other ones with added benefits, you will need to be +18 but have decent income.
I work for Yorkshire Bank and i'd recommend using your credit card for any purchases and then just paying the bill as soon as you can view it on internet banking, that way you have the cover and its interest free!
18 is the age limit for credit cards, but only if you have a good enough credit rating. In these times, it's unlikely most 18 year olds will be able to get a card, without some help. For example, I have a credit card with HSBC because my mum knows the people who work in the local branch, and so they were willing to give me a low-level (£500) card which helped me build up some credit rating. This credit rating means I could get another low-level credit card from Nationwide, but because it's a joint account with my mum, I can get one with a much higher limit, simply because she's my guarantor.