I had loads of problem opening a bank account when I first got here. I originally went to HSBC thinking it would be easier to open an account there but it was a terrible experience. To be fair to HSBC though, I think it was mostly just the person I was dealing with. She had no idea what was acceptable proof of ID, and she wasn't familiar with the different accounts, etc. She made it basically impossible for me to open ANY account! It was incredibly frustrating for me because I had started working, and as I didn't have a bank account, they cut me a check. But then no bank would cash the check without me having an account. Another catch 22.
I finally opened an account with Barclays and it was really simple. I just filled the online application form in and then had to go to my local branch with my passport and an electricity bill in my name and got a current account opened straight away. It does not have an overdraft, but I specifically asked for an account without an overdraft as to avoid any further hassle (by this point I was just incredibly desperate for a place to put my money!) but I still got the normal visa debit card.