RANIBAGH, India—In the ghostly reception area of an HMT Watches Ltd. factory, decorated with a space-age chandelier and a conversation pit—relics from a groovier era of Indian industrial optimism—the wall clock reads “8:15.” It’s actually closer to noon.

Out on the factory floor, Narayan Singh Khanayak sits at his work station, staring at another dead wall clock. He installs crowns on wristwatches, but there are none to install right now. Asked why he would be staring at the clock, he says: “You’ve got to look somewhere!”

At HMT Watch Factory V, nestled between Nepal and Tibet, time stands still in more ways than one. A half-century ago, Japanese watchmaker Citizen came to India to help HMT Ltd., a government-run machine-tool maker, start a watch subsidiary, then left.

Today the business, HMT Watches, still makes timepieces that are the height of fashion, circa the 1960s. But probably not much longer.

HMT’s classic designs sure look swell. There is the charcoal-face Pilot, with its glow-in-the-dark “12”—the better for someone like, say, a fighter pilot to read during dogfights. The whiteface Janata, with its red, whisker-thin sweep-second hand, would look snazzy on the wrist of Cary Grant in “Charade.”