The answer is quite boring: Tolkien didn’t like the idea.
Peter Jackson, who directlydirected a slightly better-known film adaptation, met Paul McCartney at the Academy Awards in 2002, where he learnt about these plans. He gave a comment to a New Zealand paper:
Jackson, whose own version of the first book in the fantasy trilogy, The Fellowship of the Ring won four Oscars this week, told the newspaper that the Beatles plan fell flat when author J.R.R. Tolkien rejected the plan. […]
“[The film] was something John was driving and J.R.R. Tolkien still had the film rights at that stage but he didn't like the idea of the Beatles doing it. So he killed it,” Jackson told the newspaper.
Matthew Schmitz has an article at First Things has written a bit more about the plans, and has a plausible guess for why Tolkien felt animosity towards the Beatles:
In a 1964 letter to Christopher Bretherton, Tolkien complained about “radio, tele, dogs, scooters, buzzbikes, and cars of all sizes but the smallest” making noise “from early morn to about 2 a.m.”
“In addition,” Tolkien wrote, “in a house three doors away dwells a member of a group of young men who are evidently aiming to turn themselves into a Beatle Group. On days when it falls to his turn to have a practice session the noise is indescribable.”