Getting into an Oxford/Cambridge grad program with only an undergraduate degree from a US school is incredibly difficult...most international students who undertake these programs already have a Masters in their field. Why? Because the OxBridge undergrad consists of studying one's major field exclusively...so, while I, as a philosophy/psych major from U of Michigan, have studied perhaps 3 full semesters of philosophy, 3 of psych, and 2 of 'other' (courses not related to either major) my equivalent student in England has had three straight years of intensive phil or psych.
However, it's not entirely impossible. Had I wanted to go for my PhD Phil I could have been taken under a tutor I worked extensively with during my year abroad at Oxford...and my best friend from my exchange program just started her PhD in History of Science at Oxford, fresh out of undergrad (she also befriended a professor there, a guy who is apparently in the very top of the field!).
If you weren't set on PhD programs, Oxford has a litany of one-year courses, similar to masters, in a variety of fields that aren't nearly as exclusive. I remember meeting a few people who graduated from Y and H law schools and then went to Oxford to get a degree in International Relations/LLM kind of stuff.