That one doesn't sound familiar but it reminded me of these:

I had a little dog
His name was Tiny Tim
Put him in the bathtub to see if he could swim
Drank all the water
Ate all the soap
Had to phone the doctor before he choked
In came the doctor
In came the nurse
In came the lady with the big fat purse
Out went the doctor
Out went the nurse
Out went the lady with the big fat purse.

Down the Mississippi
Where the boats go Push

I had a little car
1948
Went around a corner
And I slammed on the brakes

Policeman policeman
Do your duty
Here comes (insert name)
The bathing beauty
She can do the rhumba
She can do the splits
She can lift her dress
Right over her hips.

On a mountain stands a lady
Who she is I do not know
I'll go ask her hand in marriage
She must answer: (pepper - for those that don't remmeber really fast)
Tinker, tailor, soldier, sailor
Richman, poorman, beggerman, thief
Doctor, lawyer, Indian chief

Miss Polly had a dolly who was sick sick sick
So she called for the doctor to come quick quick quick
The doctor came with his bag and his hat
And he knocked at the door with a rat tat tat
He looked at the dolly and he shook his head
And he said Miss Polly put the dolly to bed
He sent her to the store for a pill pill pill
Saying I'll be back in the morning with the bill bill bill

From the days in Canada before Medicare.