Applying to Oxford and Cambridge vs. the U.S

What are the differences between the UK and the US in terms of what an applicant needs to do to get in? What sort of profile would make one a ‘safe bet’ for admission to, say, Oxford or Cambridge?

There is no ‘safe bet’ for admission to Oxford or Cambridge because it contains an interview component. If you screw up the interview or have no idea how to answer the interview questions, then you will not get in even if you receive 6 A*s in A levels or 45 in IB. Your grades, personal statement and admission test result will get you the interview, but from there on, it’s dependent on your performance in the interview. For example - I had a student who was predicted 39 in IB but gained admission into the top law school - Oxford Law, and yet other students who received 44 or 45 had gotten rejected from Oxford Law straight after the interview.
I would say if you are preparing for the US, you are indirectly preparing for the UK. Since what the UK requires is encompassed in what the US requires. There is a difference in admission exams and interview though - US requires ACT/SAT, UK requires different exams depending the major (eg. TSA, LNAT, BMAT, etc.); the interviews are more academic in the UK and more personal in the US.