Technically the Queen is still sovereign ruler, but all the power is with parliament and she's no more than a figurehead. The queens speech (well, the policy element) gets written for her by the government and she just adds in the personal and family elements (or her speechwriter does)
I think that's much the same in the church of england. Technically she's the top bod, but I don't think she ever exercises that power to actually do anything.
The royal family are in such a precarious political and economic position, and so reliant on public good will to stay on their gravy train, that they really don't have the power to make even the smallest of waves unless theyre supporting an extremely populist view (and supporting an extremely populist view isn't going to make too many waves anwyays)
But as Symmetry just pointed out she has no authority over the catholic religion