There is nothing which would disqualify you from getting a place, but since the admission decision is entirely dependent on the interview, it is impossible to say whether you would get in. It is reasonable to assume that your chances are likely better than your chances of getting into Stanford, assuming you don’t have any major hooks, and worse than your chances of getting into Berkeley L&S. However if you give the impression that Oxbridge is a backup to top US schools (for example by saying you’d like to become a doctor), then that would be viewed negatively (i.e. they really hate to be turned down). Remember that the yield for domestic UK applicants who meet their offer is something like 99%.
And read the statistics. They state the proportion of applicants each year who were admitted by subject and TSA score. Indeed looking on The Student Room you can find the full spreadsheets of admissions data by candidate (anonymized) released under the UK’s freedom of information act.