@TTH,
Thanks for the links. I, too have searched for this old song, and was never able to find it. My mother's side of the family was French/creole, and I see that the song is labelled as both Creole & African-american. Her version, also sung to me back on the early 50s, used the 'loola-loola' variant. I see some words in her version were closer to the very oldest sheet music, but not all.
Here is her version, which I assume she learned from her own mother, who was born in the 1900s:
Oh, Ta loola loola loola loola bye bye
You can have the moon to play with
And the stars to run away with
If only you don't cry.
Oh, Ta loola loola loola loola bye bye
In your mama's arms she'll rock you,
And once again we'll sing this song,
Oh Ta loola loola loola loola bye (this line descended in pitch)
Oh, Ta loola loola loola loola bye (this line ascended, ending quite high).
I sang this part, which I realize now is only the chorus, to my own two children. The full song, once I heard it, did sound oddly familiar, but I suspect mother sang the full version only rarely, and forgot much of it as time went by.