You could try Nationwide Building Society. While I don't have the credit issues you seem to have, in my experience they're the most people-friendly of the big banking institutions and might be more understanding of your problems.
Do you want a debit card or a credit card? - Natwest do a Step Account that gives you a debit card and DD facility, but no overdraft. For credit Cards, go to Capital one or Vanquis. The two credit card companies are an absolute rip off though - only get one if you are sure that you are able to pay the balance every month as the rates are really high - also look out for the fact that they try and get you to make late payments - theey nevare make it easy to pay.
Hope this helps.
Halifax do a Basic Bank Account that provides you with a visa electron debit card. They also allow standing orders/direct debits on the account and online banking. But as with the above mentioned Natwest step account they do not issue over drafts on the account