I too am interested about the origins of this song, since it lends itself to a dance, which can get everyone onto the floor without needing a partner, if you have mixed ages.
My songbook 'Apuskiduski' calls it 'American traditional'. I wasn't know to me as a child ('40s), but all my children sang it in Playgroup. So I would say it's trad. over here too, now.

My dance version starts in a circle, holding hands (except when doing the 'action'). Sing.
All to the middle...Clap/Stamp/Nod/Shout
All out...Clap/Stamp/Nod/Shout
Circle left...Clap/Stamp/Nod/Shout

So how old is it? When do you first remember singing/hearing it.

I did try searching, but got alot of parodies (which may itself show how widespread and traditional it is).