The book never names the queen, but the chronology, corgis, and mention of Diana and other people make it clear that it's her. The story starts when she's out with the corgis and they go bounding into a mobile library near the kitchen. Once she's got them out, she feels she must borrow a book -- bound by duty. She asks for recommendations from the two inside -- the librarian and a worker from her kitchen named Norman -- and takes a book with her. She dutifully reads it but finds it dull. She takes the book back, sees Norman again, and he recommends a book by a gay author (being gay himself, that's his preference in books). She also gets to keep the first book because it was on a Free shelf. She likes the new book and soon enough Norman is installed as a special page who may sit and read in a chair outside her office when she doesn't need him.
They read many books, hers mostly non-fiction, and talk about them and as this happens, she becomes less dutiful. Her reading is occupying time when she should be opening factories and going to concerts. Her secretary arranges to send Norman to a distant university, all expenses paid, and then the queen starts using the lady-in-waiting to get her books. (The queen is not sufficiently interested in what happened to Norman -- still too dutiful.) The queen starts making notes in notebooks, not just of what she's reading, but what she thinks about them. She's surprised when the prime minister doesn't know the history of the Middle East, because she's been reading about that, and surely it's part of his job. She lacks a book one day and rereads the free one from the mobile library. Now it's fresh, funny, interesting! She realizes that she has learned to appreciate and understand books.
Since she's still reading, the secretary goes to find an ancient servant of hers and gets him to talk to her about not reading. The servant is so old he keeps going to sleep and leaking a bit, but when she talks to him about the memoirs he's writing, he perks up and suggests Her Majesty write! His mission fulfilled, he's taken back to a little cottage and his dotage. The queen had already thought about writing, but this kicked her over into doing it. She hides her notebooks from the staff. The staff believes that because she isn't as careful about not wearing specific clothes too frequently (one has a large wardrobe), she's becoming senile.
During a dutiful visit to a college, she's served by Norman. She finds out that he's a graduate student, and what her secretary did to get rid of him. Norman is back as an assistant and the secretary is gone. She assembles the privy council for tea and talks to them about things that worry them and then utters the line that made me crow a bit. It's the last para of the book, and I'll put it in rot-13. If you're not used to rot-13, go to http://www.rot-13.com and copy the below in and it will translate.
I rather like that it's an alternate universe story instead of it being a way to make fun of the queen and her duties. If you're likely to read it for the pinches at Elizabeth, you'll probably find it more funny than I did. If you're likely to read it as an alternate universe, well, it seems quite reasonable.