We used to sing and play this game when I was going to Primary School in New Zealand adout 1947.
We would choose a person to be Punchinello, form a circle around him/her and sing
"What can you do Punchinello Funny Fellow
What can you do Puchinello Funny Man"
Punchinello then does some action which everyone follows singing
"We'll do it too Punchinello Funny Fellow
"We'll do it too Puncinello Funny Man"
then
"Who do you choose Punchinello Funny Fellow
"Who do you choose Punchinell Funny Man"
This may have been brought from America by the American Servicemen
who were in New Zealand in the early part of WW11