Yes it is what it's about isn't it.
My husband (American born and raised) points out that this is very much like the bawdy song often called "Gang-bang Lulu" (sorry to lower the tone of this thread)
Gang-bang Lulu has a verse that goes:

Lulu had a baby his name was Tiny Tim
She flushed him down the toilet
To see if he could swim.

Oh! Gang-bang Lulu! bangin' around all day
Who we gonna bang on
When Lulu goes away?

The tune is the one used for many Hobo/bawdy songs and it fits "Miss Lucy had a Baby" very well too.

This takes us away from the aligater purse a bit. Cheers, Joy