The Queen is the monarch, which is why she and the royal family are expected to remain neutral. So, this directly implies that she has to forfeit her right to vote. Since they cannot vote, members of the royal family aren’t allowed to hold any type of political office either. Furthermore, they aren’t even allowed to express any political views in public.
While celebrities are okay when you approach them for an autograph or a selfie, the royals wouldn’t oblige. Though they don’t take selfies or give autographs, Kate is known to pose for selfies in the past. However, even the Duchess of Cambridge would decline politely if you ask for her autograph. In fact, the royals aren’t allowed to do so because of the risk of it being forged.