The words of the Hokey Cokey are generally ascribed to the great lyricist Jimmy Kennedy [1902-1984], who wrote words of such all-time greats as Red Sails in the Sunset, South of the Border, Hang Out the Washing on the Siegfried Line, Teddy Bears Picnic, & dozens more [see Wiki].

But in case of Hokey Cokey, less original than usual, as they are pretty well identical to verses of subject of this thread. Whether orig UK or US I offer no opinions; but can definitely take it right back to my pre-WWii London childhood. We would often do it as a round dance at Woodstock School, Golders Green, N London, which I attended aged 5-7, 1937-39. Standing in a circle holding hands, we would progress in & out line-by-line singing the chorus

Here we go Looby-Loo
Here we go Looby-Light
Here we go Looby-Loo
Upon a Saturday night.

Then, breaking hold with appropriate actions

You put your right hand in
You put your right hand out
You put your right hand in
And shake it all about

Rejoin hands for chorus, & then etc.

I am absolutely confident of accuracy of these recollections and the dates I have given.