Interesting thread with all sorts of suggestions. Going back to the OP I would say the first rhyme offered is the most likely fit to what the OP asked for.

Miss *olly had a dolly who..... and then the 3 repeats certainly fits the well-known song. Having said well-known I can't find it under any of the most likely titles in my extensive children's rhyme index.

Anyway here's what I can remember.

Miss Polly had a dolly who was sick, sick, sick
She sent for the doctor to be quick quick quick
The doctor came with his bag and his hat
And he knocked on the door with a rat a tat tat.
He looked at Miss Polly and he shook his head
And told Miss Polly put her into bed
He wrote on his paper for a pill. pill, pill,
I'll be back in the morning with my bill, bill, bill.

The tune has some affinity with the dance tune 'The Morpeth Rant' and also the theme to The South Bank Show (UK TV)