The only recording I can find of this song is by Ivan Stiles, on his album Promises Kept. He calls it WHIZZ FISH SONG.

The San Joaquin Valley Library System, which has apparently indexed all the songs in several California libraries, says that WHIZZ-FISH appears in a book called "Song Fest" written (or edited?) by Dick Best, and published in 1948. The Fresno County Free Library has a copy.

(That last link takes you to the search page for the index. It looks promising and I plan to bookmark it.)

THE WHIZZ FISH also appears in a "fake book" called Children's Song Favorites, (2003).