The correct title of the above-posted song is NUTTIN' FOR CHRISTMAS, and it was written by Sid Tepper and Roy C. Bennett. It was first recorded in 1955 by Art Mooney & His Orchestra, with vocalist Barry Gordon—a 7-year-old one-hit wonder who had first appeared on Ted Mack's Amateur Hour. It has also been recorded by the Fontane Sisters, Stan Freberg, Spike Jones, Rosie O'Donnell, and others. A couple of Dr. Demento anthologies include the version by Stan Freberg. This info comes from AMG - All Music Guide.

I can't tell whose voice you hear at the site LJC linked to.

The version that Homer & Jethro recorded has different lyrics. It goes:

First line of verse: "Johnny is a little brat. I'm gonna tan his hide."
First line of chorus: "He's gettin' nuttin' for Christmas."

I would be thankful if someone would post H&J's complete lyrics. I have been unable to find them online.