On the Oxbridge application form, they do not ask for your A levels. However, for a competitive doctorate, they may not like the fact that you have only a 2:1. If your distinction is from a highly regarded university, this will REALLY help. All universities are NOT equal in the eyes of Oxford and Cambridge. Also, with regard to quality, no, the only grad. school that even comes close is LSE (perhaps Imperial) the other institutions have nowhere near the research reputations of Oxbridge-go take a took at the RAE.
Best of Luck.