Hi Averroes,
I'm not sure specifically for religion or theology, but most PhDs are, like you said, organized under a supervisor/supervisors (at Oxford that can be one or more supervisor, if necessary).
I'm in the Social Sciences and haven't started the PhD portion of my degree yet (I'm still in the M.Phil), but in our department there is also more to it than that. Lectures, for one— they aren't required, of course, but there are different lectures (meaning lecture-based courses, usually required for Masters students) almost every day that PhDs often attend. Language study, technical courses, and research method courses or trainings also might take up time on a weekly basis.
It also may depend on your supervisor. Some supervisors structure their "supervision" more like a course in that they may have specific assignments, requirements, and writing pieces spread across different portions of the year. That being said, some go in the complete other direction and may not even organize a meeting if you don't take the initiative to do it.
My personal opinion on it is the D.Phil (not a PhD, technically, at Oxford— just a reminder), is very much about what you make of it. If you want to be completely independent and rely on your supervisor just for direction, that's entirely possible. But, of course, at a world-class institution you have a great opportunity to take advantage of courses, lectures, and other resources. This goes for the libraries, as well as your college (which is a part of the experience that is often overlooked by US students). Also, you have the best libraries!
That being said, to be successful at the D.Phil level you have to be the kind of person who is assertive enough to work to arrange their experience as best for them. As one of my advisors explained to me, there is absolutely no "hand holding" in UK doctorate programs. The expectation is you come in knowing what you want to do and what you'll need— then you go after those things.
I hope this helps. The best advice I can give is to download a D.Phil handbook from Oxford's website for your specific course. The experience can vary a lot department-to-department, and the best way to see what's required of you is to look at the official handbook— they should give you a picture of what you can expect.