I found this in the book "Shimmy Shimmy Coke-Ca-Pop! A Collection of City Childrens Street Games and Rhymes" by John and Carol Langstaff, published 1973.
Sung to the tune of "Twinkle Twinkle Little Star":
Ringo, Ringo, Ringo Starr How I wonder what you are Underneath that mop of hair Ringo, are you really there? Ringo, Ringo, Ringo Starr How I wonder what you are
You're so vain, you probably think this post is about you.