Yes, it is possible.
However, as others have mentioned, you would be applying as an overseas student, probably even if you have EU or UK dual citizenship. This would entail ~$50,000 per year out of pocket expenses with no aid available. Secondly, you would have difficulties applying for US residencies. Again, not impossible, but you would have to obtain ECFMG certification and probably arrange rotations as a visiting med student prior to applying for US residencies. I believe someone from Oxford entered Johns Hopkin's internal medicine residency program within the last decade, so it is possible (I would suggest doing a google search for Oxford or Cambridge MBBS graduates in US residency programs for more insights). However, if you plan to practice in the UK or in a Commonwealth country, this is a different story - it would probably be to your advantage, though there would be citizenship issues.
In terms of gaining admissions, you would have to take the BMAT (or similar) as well as submit A-levels equivalent exams (IB exams work well; AP exams can be used in Oxbridge from what I have seen, but you would need something like 8 AP exams with 5's on them, including all of the science ones - Phys, Chem, Bio). Also, you would have to go through the interview process - which, in my opinion, is more stringent than that for US allopathic schools. These interviews, especially at Oxbridge, will involve answering technical or scientific questions, some of which are designed to be above your knowledge level. For example, you may be shown a graph of heart volume vs. pressure, be asked to identify which parts of the contraction cycle each part of the graph represents, be asked to identify what the area of the graph represents, and then explain possible pathological conditions in aberrant graphs. Again, this is reflective of a mode of thinking that in my opinion is above even AP level skills in US high schools.
Lastly, US and UK/EU/Commonwealth health systems differ in many regards - not just coverage rates and patient waiting times, but also in limits on patient care expenditures, uses of certain pharmacological entities, availability of stem-cell based therapeutics, malpractice & tort systems, retirement age, and physician compensation. It would behoove you to investigate these thoroughly, not just before matriculating in the UK, but before even attending an interview in the UK. If you are applying from the US, most British interviewers will want to know that you have a detailed appreciation of their healthcare system, and that you can explain your motivation for wanting to study/practice medicine there as opposed to in the US.