Incidentally, the tune is common and used for a couple of children's songs and games. In a 2015 message, Amy Creech gave a link to the Seagreen Singers; they gave another name "Stand and Face Your Lover" The link Amy used isn't working for me, but I looked at this webpage: https://www.seagreensingers.com/music/WilliamBrownV1-5.htm

I didn't recognise the name "Stand and face your lover", but it's the singing game "In and out the windows" which has already been mentioned. George Formby used the lines and tune for the intro and ending of "Let's All Go to Reno", and also intersperses "Have you seen the dairyman" (aka Have you seen the muffin man" and London Bridge is Falling Down.