The snap, crackle and pop of vinyl marks the start of The Ballad of Wallis Island, a music-themed rom-com that’s half-whimsical com and half-unrequited rom. In the absence of a new film by John Carney, who has the template nailed down by now, audiences at Sundance lapped it up. Outside a festival atmosphere, however, its commercial prospects aren’t immediately clear, since the humor is exceedingly British and the references often seem kind of dated (perhaps a result of the film’s 18 years in development after starting life as a BAFTA-winning short in 2007).

The record being played is an album called Way Back When by McGwyer Mortimer, a folk duo who hit their peak in 2014, and the song — which tells “a story about a man who’s tired of life” — signals the arrival of one half of the band, Herb McGwyer (Tom Basden), at a remote island. The island...