The Levy Sheet Music collection has an undated song by Harry Dacre called Playmates, and other sources show a publication date of 1899 for that song. The chorus starts, "Playmates were we! Little we thought it then...," so that's not the song we're talking about here. anybody have time to transcribe the Dacre song?
Apparently, the song above and in the Digital Tradition was written by Saxie Dowell and published in 1940. It was one of those songs my little sister used to sing to aggravate me.
-Joe Offer-