I can't speak for Cambridge and I've never been a student at Oxford, but I have worked in the Graduate Admissions Office at Oxford. When I worked there - granted a few years ago - it wasn't the snobby entrance requirements you'd think - usually if a professor or a faculty was interested in you and you had at least a reasonable first degree, you'd probably be given some sort of opportunity.
My first degree is from an old teacher training college (decades ago!), my first MA is from a middle-of-the-road American college, my second MRes will be from a lower tier British university, but I'm still considering applying to Oxford. To be honest, I have no idea if I'll get in or not (I already have a PhD place somewhere else), and the only reason I'm holding back is that I have a family attached to me and I can't afford Oxford house prices.
But you've got nothing to lose and everything to gain and if you are interested in Oxford or Cambridge, just apply.