One of the companies doing this is Think Banking but there are others. However, contrary to what you may think from their site, they are NOT part of any bank. They are an independent company and they just put all your money into an account with everyone elses (which is held at a bank, but the bank named does not control that account).

I'm not suggesting that there's anything dodgy about them. It is an expensive way of getting an account - you don't even manage your own money and you only get a prepaid card loaded with money each month. There are better ways of getting an account in your own name without credit scoring.