In a time where the public has a more complicated relationship with celebrity than ever, and more male stars are outed for their indiscretions, what does the future hold for a brand so dependent on famous people as Madame Tussauds? Vloggers, reality stars and even Grumpy Cat — whose figure is terrifyingly animatronic — have their own waxworks at the attractions now.

“If there’s a massive celeb and it’s big news worldwide, that’s someone we’d probably need to look at and do,” Ms. Haughney said.

Fame moves quickly and is never certain. Can the melted wax keep up?

Get Me Bodied

While Tussauds is the largest network of waxworks museums in the world, it hardly holds a monopoly: in Baltimore, there is the Great Blacks in Wax Museum; in Paris, Montreal, Prague and Seoul, the Musee Grevin; the Heroes of Baseball Wax Museum is in Cooperstown, N.Y.

A handful of notable wax museums have closed in recent years — in Buena Park, Calif., the Movieland Wax Museum (which was previously the largest wax museum the United States) in 2005; Ontario’s Criminals Hall of Fame Wax Museum in 2014. Yet, Tussauds’ most recent attraction opened as recently as last year, in New Delhi.

“Waxworks have this simultaneous lifelikeness and deathlikeness,” Michelle Bloom, a professor of comparative literature and French at University of California, Riverside and the author of “Waxworks: A Cultural Obsession” (2003), said in a phone interview. “But a wax figure can pass for a living person at the same time that it’s very corpse-like. I think we’re drawn to both, really.” In Dr. Bloom’s book, she explores the cultural phenomenon of waxworks through art, film and literature.

“Their lifelikeness and yet — they’re not real! — means that we can do what we want with the figures,” she said. “A lot of Hollywood horror films and works of literature get to this perverseness. There is usually a male character who is enamored of or obsessed with the wax figure of a woman. It’s the ‘ideal woman’ who doesn’t talk and is literally malleable. He can control her. He can project his fantasy onto her.”