Michael Rosen does a performance and has written a children's book.

He has a blog where he mentions other versions he's heard himself or has been told about :

"This puts one origin for the song/rhyme as early as 1954"
'I was a brownie and girl guide about 70 years ago - in those days, we sat in a circle and did the actions to ‘I’m going on a lion hunt’. I definitely remember how we all screamed ‘a lion’ when we found ‘something soft, something furry’ in the cave and how we scrabbled our way back home, panting as we arrived!'
Margaret E. Sandercock - Facebook; April 8 2024

no date - Brownies (Girl Guide movement)
but their version was a 'Lion Hunt'

1983 - Linda Goss
,,, with obstacles of a tree, a river, a cornfield, a 'lah dee dee da dee' and a cave.
,,,mostly the moves through the obstacles are done with noises not words. The repeated line is 'can't go over it, can't go around it,' till it gets to the cornfield which they go 'through'. At the cave, the singer takes you to 'feel something' and after they realise it's a bear, they run home (no words, just noises made by slapping legs), and the last line is 'we were lucky that time'.
https://youtu.be/lY_-mA9AjPQ?t=11s

1973 - Sandra Stroner Sivulich,
illus by Glen Rounds, published by Dutton
'the obstacles are trees, rivers and caves'
Thanks to Jo Brodie on Twitter/X

1977 - Kathleen Savage and Margaret Stewart,
illus. by Leonard Shortall, published by Prentice Hall
'the obstacles are a river, bridge, cliff and a swamp'
Thanks to Jo Brodie on Twitter/X

1977 - Alison McMorland - album 'Funny Family'
American version ,,, devised in US summer camps. Some of these included a repeated line about guns
[That was the first version that Michael heard.]
http://www.alisonmcmorland.com/music/the-funny-family/09-were-going-on-a-bear-hunt.mp3
on
http://www.alisonmcmorland.com/player.asp

1986\87\89 - Michael Rosen
grass river mud forest snowstorm
illus Helen Oxenbury, published by Walker Books.
www.walker.co.uk/9780744523232/were-going-on-a-bear-hunt/
- interesting video there of Michael talking about the creation of the book.
performance : https://youtu.be/ytc0U2WAz4s

more :
https://michaelrosenblog.blogspot.com/2024/04/the-true-story-of-making-of-book-of.html