Well, this is absolutely marvellous!
I've been hunting around for years, on and off, to find this song which was a staple of ABC morning radio when I was a child in the mid-1950s. I loved it and remembered some of the lyrics but didn't know the title nor the performers.
Russ Tyson played it almost every morning for what seemed like years from mid-1950s to the early 1960s.. I can't remember when it finally drifted off the play-lists.
From Leeneia's mention of You-Tube, I looked it up and found this scratchy, but reasonable, recording from the 78rpm record, and it IS exactly what I remembered and expected.

https://www.youtube.com/watch?v=jyQWhp310tI

Thanks to you all for helping me to find a childhood memory that was special to me.