Possibly they decided to pass on The Lord of the Rings because it was written prior to the cut off date (though published afterwards). There are seventy books on the list, split up into ten for each decade of Elizabeth II's reign - whilst The Lord of the Rings was published in the earliest 1952-1961 decade, it was written in stages between 1937 and 1949, which was before she came to the throne.
It also seems they wanted a selection of books from around the whole Commonwealth. By not including LotR in the 50s decade they were able to check Nigeria, Jamaica, Barbados, Guyana, Trinidad and Tobago, India and Wales off the list. There are currently nine other books already that are solely from England.
Finally, I think they also wanted to skew towards lesser known authors where possible - Isobel Hunter, chief executive of Libraries Connected, said the campaign will help “libraries to introduce readers to writers they might not have come across before". Pretty much everyone is already well aware of Lord of the Rings, so they might have preferred to free up space for some more obscure books and authors (though I note some very well known books like Douglas Adams' The Hitchhiker's Guide to the Galaxy are on the list).
You'd probably need an interview or quote from someone on the selection panel for a true answer, though. There is a mention of why Harry Potter ultimately didn't make the cut:
Susheila Nasta, professor emeritus of modern literature at Queen Mary and Westfield University, said there was a “great discussion about JK Rowling” before the list was completed.
She added: “She was long-listed with Harry Potter and the Philosopher’s Stone.
Space has been made available for someone who is equally good, but whose work was not so well known. There were some very tough decisions.’
The website does claim "The books on The Big Jubilee Read list have been chosen for adults", so I'd be a little surprised to see Harry Potter on it anyway.
