From a very recent biography of Tolkien by Raymond Edwards:
"In the late 1960s, the Beatles were keen to make a version of The Lord of the Rings, with the four of them playing Gollum, Frodo, Sam, and Gandalf. Tolkien, who detested the group as a whole, and the bumptious John Lennon in particular, was furiously opposed; they did not secure the rights."
I am guessing:
Gandalf - George
Frodo - Paul
Ringo - Sam
John - Gollum
Paul I am sure would have insisted on being Frodo, and really George has to be Gandalf. The other two are tough ones.
Yoko could have been Wormtongue.