Troughton. The Quail's Egg

The Quail's Egg is a children's book by Joanna Troughton, with art by the author, published in 1988. It is available at the Internet Archive.

Troughton indicates that the story comes from Sri Lanka, but she does not give her source; I'm 99% sure that she was relying on Parker's Village Folk-Tales of Ceylon, "The Female Quail," which is available at the Internet Archive. A quail's egg rolled into a rock, and she needed help to retrieve it. The mason refused to help, as did the village headman, and also the pig, the vedda (forest dweller), the timbol thorn, the fire, the water-pot, the elephant, the rat, and finally the cat who said yes, so the final cumulative chain is: "Well then, the Cat went to catch the Rat, the Rat went to creep into the ear of the Elephant, the Elephant went to make muddy the Water-pot, the Water-pot went to quench the Fire, the Fire went to burn the Timbola, the Timbola went to prick the body of the Vedda, the Vedda went to shoot the Pig, the Pig went to feed in the rice field of the Village Headman, the Village Headman went to tie up the house-door of the Mason, the Mason went to cut the rock, and take and give the egg." Parker helpfully cites some other versions of the story.

In her preface to the story, Troughton compares the story to the English folktale of the old woman and her pig, which is indeed the most well-known "chain of command" stories. Troughton has simplified the chain in her version, leaving out the vedda and the headman; the mason is the only human being in her chain, and she says "mouse" instead of "rat." Here you can see the cat, the mouse, the elephant, the water, the fire, the creeper, the pig, the mason, and then the quail... with a newly hatched quail chick!

I like how the cover inside papers show all the characters!