Here are the results of my research (which is inconclusive):

The following songs might be relevant, but I have no lyrics to confirm this:

A WONDERFUL TIME recorded by
Deal Family, 1927

WE'RE HAVING A WONDERFUL TIME
Written by Irving Bibo and Don George
Sung by Jack Oakie, Donald O'Connor and Peggy Ryan in the film "The Merry Monahans" (1944)

HAVING A WONDERFUL TIME (WISH YOU WERE HERE), written by William Livingstone, and recorded by -
Tommy Dorsey & Clambake 7 in 1937
Dinah Shore – date unknown but probably around 1947 or 48
Benny Goodman & His Orchestra in 1949

HAVING WONDERFUL TIME recorded by
Jolly Coburn & His Orchestra, 1937
--which might be related to the film "Having Wonderful Time" (1938) starring Ginger Rogers & Douglas Fairbanks, Jr.
(However, IMDb.com lists 4 songs in the soundtrack of this film, and none of them has this title.)

I think we can eliminate the following songs. Although they contain the phrase "having a wonderful time," they do not contain anything like the other lines quoted above:

Title: OOH LA, LA (I'M HAVING A WONDERFUL TIME)
First line of verse: I just got a note that my brother wrote
First line of chorus: Ooh, la, la, when I start kissing
Creators: Green, Bud [lyricist]: Nelson, Ed. [composer]
Publisher: New York: A. J. Stasny Music Co.
Date: 1918
Sheet music viewable at The Archive of Popular Music.

Title: NOW AND THEN BUT NOT ALL THE TIME
From the musical production: "Oh, My Dear!"
First line of verse: I've been having a wonderful time you know
First line of chorus: Now and then we must have jolly time, now and then we all take a little wine
Creators: Bryan, Alfred [lyricist]: Schwartz, Jean [composer]
Publisher: New York: Jerome H. Remick & Co.
Date: 1919
Sheet music viewable at The Archive of Popular Music.

Title: I'M GOING 'WAY BACK HOME AND HAVE A WONDERFUL TIME
Composer: Schwartz, Jean
Lyricist: Jerome, William, 1856-1932
Performer: Marino Sisters
Publisher: Waterson, Berlin & Snyder Co.
Place of publication: New York
Date of publication: 1916
First Line: Call a taxi, please, me for shady trees, I've got that hometown
Chorus First Line: I'm going 'way back home and have a wonderful time,
Artist: Barbelle
Sheet music viewable at the Indiana University Sheet Music Collections.