“Love Actually” director Richard Curtis and songwriter and musician Ed Sheeran go way back: Sheeran’s first girlfriend even babysat Curtis’ children.

Since then, they’ve collaborated on several projects, including Comic Relief and the Danny Boyle-helmed 2017 film “Yesterday,” in which Sheeran stars and Curtis co-wrote. So when Curtis was conceiving his first animated project “That Christmas,” he showed Sheeran some early storyboards, and the original conversation was that it would be a musical.

“He asked if I wanted to write the music for it, and I said, ‘Cool,’” Sheeran says from his home in London. “There was one scene, and I wrote a chorus for it, but didn’t hear anything back.’”

Two years later, Curtis asked Sheeran to finish the song “Under the Tree” and updated him on the story. “That Christmas” which is now streaming on Netflix, would no longer be an animated musical, but rather...