12 if you're native 24 if you're a freeloader from the middleastern regions
It depends on what kind of account.
If it's a savings account, it could be any age. My parents did it for me so if relatives sent a cheque as a present for my birthday or Christmas, I had an account to pay it into. And I learned at a young age what a cheque is and what to do with it. All great stuff. The bank is totally OK with that because going into the bank is the only way to get money out. Or with online banking, the bank can easily say no to any request to take more out than you actually have.
But if you want an account with a debit card so you can use that to spend with, the youngest I know is HSBC, who will let you have one at the age of 11. If you're under 16, a parent will have to go to the bank with you to open the account and sign up to take responsibility for you.
Yes, I know banks offer ways to open an account online but there is really no point in this if you don't have one already. They need to SEE proof of ID and proof of your address, so it's best to make an appointment to do it, and then you can take the documents they need to see and show them, and get the account opened all in one go. Plus once you're there, you can ask anything you like about how it all works. Been there, done it… they might not have anyone free who can do this for you if you just walk in, so it's best to make an appointment.
I learned my lesson when I opened an ISA with Nationwide Building Society to put some savings in. I'd had a bank account for years so I tried opening it online, but they STILL insisted that I went in with my proof of ID and address so they could see it for themselves. And I'm a LOT older than you (I'm 51)! (I have to say that was a bit unusual – I also have money in a bank that only operates online, and they never asked – I told them what bank my current account is in, and they just asked that bank for a reference.)
If you want a "proper" current account, where you also have a cheque book and can go into overdraft, you have to be 18. The reason for that is that if you borrow money (by writing a cheque for more than is in the account, or just spending more than you have with your debit card), then if you aren't a legal adult, they can't sue you to get the money back if you don't repay it. And you're not a legal adult until you're 18.
Any age in the UK, but a parent will have to go with you with ID.
Any age. A baby can have a bank account, someone else can open it for them. The account can be converted as they get older. What is it you want to do though? No borrowing until you are 18 but there is no age on simply having an account to put money in. When I was 12 I opened a bank account myself with no help.
Edit: Clive is wrong. They can pursue someone under 18 who has an unauthorised overdraft, because they didn't give them permission to go overdrawn. I had a debit card and cheque book before I was 18 and sometimes went overdrawn. They are not allowed to give credit to under 18's, which means authorised borrowing.