In an awards season where the race still seems to be fairly wide open, one film is gaining more traction than most as the finish line wobbles into view over the horizon.

“Emilia Perez” wasn’t talked about as an Oscar hopeful after it first bowed at the Cannes Film Festival, but eight months on and Jacques Audiard’s musical thriller about a Mexican cartel boss who undergoes gender affirmation surgery to live as a woman looks like it’s in pole position to land some major statuettes.

On Sunday, it won big at the Golden Globes, emerging ahead with four gongs, including best motion picture — musical or comedy, best supporting actress for Zoe Saldaña, best motion picture — non-English language and best original song. Just two days earlier, it was the top film out of the BAFTA longlists, with 15 slots across the various categories.

The emergence of “Emilia Perez” as...