If you have a lousy Credit Score would a bank automatically turn you down even though you wouldn't want an overdraft and only use the money that you put in? Also, if you had a Savings Account with the same bank and they decided they didn't want you as a customer, would they close down your Savings Account?
I very much doubt that the status of a savings account would be adversely affected by an unsuccessful application for a current account.
All High Street banks offer 'basic' accounts to people with people with poor credit ratings. A couple of banks (Cooperative and Barclays) even offer accounts to undischarged bankrupts). So you should have no problem in finding a suitable account: http://www.moneymadeclear.fsa.gov.uk/pdfs/bank _accounts.pdf