Smart people found what I couldn't.
When I posted about the Shirley Temple Black story time song, I wrote that I couldn't find it. Shortly thereafter, ziniania's corner posted a link ↓ of Ella Fitzgerald doing a version of that very song.
I don't know Ziniania's corner, and when I clicked, I found that she is one who has a Blogger account but doesn't blog. But she (I assume) found my post and also found ↑ the video. So sorry that I accidentally deleted her comment. These things happen, especially on some of those very early morns after not a whole lot of sleep.
Then, Kay posted a recording of the actual Ms Black. The tune is as I remember, and some of the words are as well although they don't all seem to have been in quite the sequence that I posted.
Thanks to both commenters. I was pleased to hear Fitzgerald's original (I presume) version and especially pleased to hear Shirley Temple Black singing it just as I remembered — or close enough.
Apparently however, I should hang my head because I don't know how to search on my own very well.