Around this neck of the woods the song is sung to the Auld Lang Syne tune and the chorus appeared some 20 years ago on a CD of children's songs put out by Mike and Michelle Jackson.

And another song (with only one verse I've ever heard), to the same tune, I learned in the 50s;

The fish, it never cackles 'bout
its million eggs or so;
the hen is quite a diff'rent bird,
one egg and hear her crow.
The fish we spurn but prize the hen,
which leads me to surmise,
"Don't hide your light but blow your horn;
it pays to advertise."

Cheers, Rowan