Timing is off, yes, I noticed that.
It's just that I wanted my info straight,
and it has taken me a long time to locate
the perpetrators, I mean, the authors
credited with this song, lyrics and music both.

The song is
Dear Santa, Have You Had the Measles.
Copyright belongs to three individuals:
Frank Kells, Sal Messina [sometime pseudonym: Al Arland], and Al Vann. Shawnee Press / ASCAP. circa 1958.
This thanks to
the Library of Congress Catalog of Copyright Entries, Published Music:
Third Series, Volume 10, Part 5A, number 1.
Washington, D. C.: Copyright Office, Library of Congress, 1957.

someone else is more than welcome to fetch the lyrics,
if you insist on knowing them...