Like any popular or famous person, they usually have people who handle all the mail, especially if it's fan mail. Leaders of countries are also unindated with tons of mail asking for favors or photos. The White House in US is always getting invitations to weddings, because they always send a notice saying thank you, but unable to attend, and the newly married couple then have a neat little trivia for their memoirs, a letter from the then president of the US about their wedding.
I'm sure that the Queen of England gets gazillions bits of mail. She is extremely well loved by many of Englanders.
No, very seriously doubt she would see it. But it would still be nice for you to send a letter hopefully praising her or telling her some nice stuff. It doesn't hurt to let them know you like them or like what they're doing. She is an older woman, so she might enjoy getting good mail, as opposed to cranky mail.
But like movie stars or music artists, they rarely will see the mail themselves unless "their people" are directed to bring certain kinds of mail to their attention. Some artists do manage to read much of their own mail, but mostly it's just hard to read thousands of letters and emails daily.
But still, it's fun to write people like that. There's been some fun stories in news such as a young girl of twelve who had thousands of autographs from many stars---political and entertainers. Just said that she was collecting signatures, and she always managed to get them from nearly everyone. But usually, when you ask for something, other than autographed picture, it's hard for celebrities to send stuff out to fans.