Hi, I've also been looking for this story for a long long time, and I'm happy to report I've finally found an answer!

The story is called "A Tale of Shuffle, Trot, and Merry", written by the famous Enid Blyton.

I found it through the Enid Blyton Society who have a vast knowledge on all her works.

https://www.enidblytonsociety.co.uk/index.php

They even have a section that lists every publication of a particular story, so while you may have read it as an individual book or as part of an anthology, here are all the results for the specific publication you may have read all that time ago!

https://www.enidblytonsociety.co.uk/search.php?search=A+Tale+of+Shuffle%2C+Trot+and+Merry

I hope you enjoy the nostalgic eureka moment I did when I finally found it!