Thanks for asking this question, Joybell.

One romantic historical novel that I read [yes, I sometimes read books like that] had as a central character a woman who was a herbalist. This was in either the 18th or 19th century American West. In one chapter of this book [whose title and author I've regretfully forgotten], the author writes about the woman grabbing specific herbs and placing them in her alligator bag, and then rushing to provide aid to a sick person.

So, if an alligator purse was a trademark for a herbalist, it makes sense that that children's rhymes lists the "lady with the alligator purse" after the doctor, and then the nurse came but [presumably] failed to help the poor baby who almost swallowed the bath tub.

But this is just an uneducated guess.

I'd love to know more about this if there is more about this to know.

:o))