Walt Disney's first idea of having each doll in it's a small world sing his or her national anthem was a bust. It sounded terrible and they were on a tight deadline. He knew he had two guys who could help- the Sherman Brothers.
Robert and Richard Sherman had originally arrived at Disney to write songs for Annette Funicello. Their wheelhouse up until then was light pop tunes, which was exactly what was needed for Walt's most popular Mouseketeer. Walt saw their potential, however, and they soon found themselves writing songs for movies, television and now attractions. Despite also working on Mary Poppins, they took time out to write a song that could be sung in multiple languages without sounding like a terrible racket. The result was it's a small world. Walt Disney loved the song so much, he changed the name of the attraction to match it.
The company and the brothers were convinced by UNICEF to not copyright the song so that anyone could sing it. It quickly became a standard. With all the elements in place, this attraction was ready to roll at the World's Fair.