As I understand it (and with more waffle than Jubal) ...
In the UK, the reigning monarch doesn't vote in elections. This is based on the constitutional principle that the monarch is neutral and does not intervene in any part of the political process. Also, from a theoretical perspective, the election is called and the MPs are summoned so that they can speak on behalf of the people to the monarch, so it would kind of inconsistent for the monarch to be involved in the selection of an MP.
Other members of the royal family are not restricted in the same way as the monarch, so I think they can vote, but I suspect they don't, on the grounds that they too should be seen to be politically neutral.