A cool 1960s vibe, an old-fashioned love theme, dramatic music befitting a lunar landing: Daniel Pemberton’s score for “Fly Me to the Moon” needed to accomplish all these and more.

Greg Berlanti’s rom-com with Scarlett Johansson as Kelly Jones, a Madison Avenue marketing genius hired to sell the moon to the American public, and Channing Tatum as Cole, the NASA director in charge of the Apollo 11 launch behind America’s first moon landing, meant Pemberton could incorporate different musical strands “from the fun aspect to the romantic couple element, and then NASA and a kind of Americana nostalgia,” says the English composer whose credits include “The Trial of the Chicago 7” and “Spider-Man: Across the Spider-Verse.”

“I was trying to capture this beautiful, melancholic nostalgia for a time that feels very distant now in America,” Pemberton adds. “One of my favorite pieces in the film is when Apollo...