Most British universities don't map cleanly onto these categories. In theory most of them (including Oxford and Cambridge) are independent educational charities. But endowments are (by US standards) small, and they're so dependent on public funding that they tend to do governments' bidding. A little less so Oxford and Cambridge, which are richer and prouder, but it's mostly a difference in degree than in kind. There's also quite a bit of direct government regulation (e.g. by the Office for Students).

There are also a few actual private universities, some non-profit (e.g. Buckingham, a free-market experiment which has survived but not thrived) and some for-profit (e.g. BPP University). But all of our major institutions are semi-public. Ish.