“The belle epoque of watch design.” That’s how Roni Madhvani describes the postwar years of 1945-65, the years when most of the highly designed, time-only watches in his collection were produced.

The collection has developed quite a reputation among watch fans, with almost 50,000 Instagram followers drawn to his images of timepieces by brands including Patek Philippe, Cartier, Audemars Piguet and Vacheron Constantin. And he seems to be as fascinated by vintage designs of merit by lesser-known watchmakers like Titus and Universal Genève as he is by those created by famed designers such as Gilbert Albert, Jacques Cartier and Gerald Genta.

So what drew Mr. Madhvani to these timepieces while many buyers — the “herd of sheep” as he called them — gravitated toward the steel sports watches that dominate the market now?

“They’re pieces of art on the wrist,” said Mr. Madhvani, 59. He said he changes his watch every day, and during an interview in London last month, he was wearing a rare gold 1962 Audemars Piguet Ref. 5182 with an asymmetrical dial that is a riff on the shape of the letter C.

“For men, it’s difficult: Women can wear jewelry or fancy clothes or whatever, whereas for men, it’s pretty much watches and suits, and I don’t wear a suit in Africa,” he said with a laugh. “A watch tells you a lot about a person. The first thing I do when I meet someone is look at them and then at their wrist.”

Mr. Madhvani was born in Uganda, but was at boarding school in Britain in 1972 when his family and thousands of other Asian Ugandans were expelled by then-President Idi Amin. It was in the late 1980s, while he was studying at the London School of Economics for a degree in economics and international relations, that Mr. Madhvani first caught the watch bug.

Walking along Bond Street one day to the university, he spotted a modern reissue of a 1940s chronograph by Baume & Mercier in the window of the multibrand retailer Watches of Switzerland.

“It was 575 pounds and for 18 months, I saved the money,” he said, adding that he must have gone to the store around 15 times, much to the annoyance of the sales staff, before he was able to finally buy the watch.

His collection began in earnest after graduation, when he returned to Uganda and was given a watch auction catalog by his Indian father-in-law, an art collector. “The whole disease started then,” he said wryly.

Mr. Madhvani initially established his own import company there, but later joined Madhvani Group, the family business that has diverse interests in Uganda, including agriculture, manufacturing and tourism.

Today, he is a company director, managing its packaging, real estate, construction and trading business in Uganda as well as the family’s educational charitable foundation and its hospitality business in India. As a result, he divides his time among Kampala, Uganda; London; and cities in India including Mumbai.

Decades ago, hampered by Uganda’s poor dial-up internet connections and slow postal service, he gradually acquired a collection of watches through dealers, predominantly relying on online watch forums as his main sources of information and developing a camaraderie with his fellow collectors.

More recently, Instagram has become a key method for tracking watches on Mr. Madhvani’s most-wanted list. “One of the reasons I spend too much time on Instagram is that people know what I collect, and it’s a dream to wake up every morning with someone saying ‘I’ve found this’ or ‘I’ve got that’,” he said.

Over the years, he has become more focused in his collecting and is particularly interested in watches whose details reveal their provenance and makers. For example, he searches for dials that have the names of both the manufacturer and the retailer, a practice popular in midcentury watchmaking.

“Mr. Madhvani has a very carefully curated, very intellectual collection,” Alexandre Ghotbi, head of watches for Europe and the Middle East at the Phillips auction house, said by phone. “It’s not about hyped watches, he’s clearly been thinking about it and searching for a long time.”

Mr. Madhvani’s double-signed pieces include a 1950s Audemars Piguet signed with Cartier New York, a 1950s Patek Philippe from Serpico y Laino in Caracas, Venezuela, and a Rolex from Cooke & Kelvey in Calcutta (now Kolkata), India. “It brings the whole watch to life,” he said. “It gives it a personality.”

For the same reason, vintage watches with personal engravings also have become a passion for him, even though many other collectors dislike such personalizations.

Instagram, he said, had been a useful source for discovering their stories, like the history behind an early 1950s Vacheron Constantin inscribed “20-7-54, Dolly.”

The jewelry collector Clive Kandel told Mr. Madhvani the men’s watch could well have belonged to a family friend. Mr. Kandel’s family knew an Austrian Jew, who after fleeing to New York when the Nazis came to power, changed his name from Adolphe to Dolly.

But his latest fascination is with the case makers who created watch parts independently, before such jobs moved into factories in the 1990s, he said. “Those case makers, the independent ones who sat at home making, are the unsung heroes for me,” Mr. Madhvani said, adding that research into their work by the likes of the Patek Philippe specialist site Collectability is leading to a growing awareness of their expertise.

He owns two examples of the sought-after mid-1950s Patek Philippe Ref. 2546, which has an elaborately curved case, each one sculpted from its own single block of gold by a maker named Markowski. For one of these, he said he paid 35,000 Swiss francs, now $39,836, at Phillips in 2017.

“It’s almost now more about the hunt for watches rather than owning them,” he said, although he regrets that the old-fashioned “gentleman’s code” of collecting has largely disappeared in the wake of global watch conglomerates, fierce competition among auction houses and the increasing number of fakes on the market.

He cited an example of the good sportsmanship he values: He once spent a decade tracking a rare Patek Philippe Ref. 2549, nicknamed the Devil’s Horn for its distinctively shaped lugs. Being in Uganda at the time meant he had to put in an absentee bid when one came up for auction in 2010 and he narrowly missed out. When the winner, the fellow collector Jason Singer, found out how passionately Mr. Madhvani had wanted the timepiece, he sold it to him for the same price he had paid.

“It was a wonderful watch but I could tell Roni was very passionate about his watches, and particularly this one,” Mr. Singer said by phone recently. “The pieces Roni collects are about passionate artistry. It’s really all about the beauty of the dial, the case and form, and he has a great eye.’

It was a courtesy that Mr. Madhvani said he tries to copy. He estimated that he receives 50 to 100 Instagram messages a day from experienced and new collectors alike, and he tries to reply to them all, often recommending that enthusiasts look for beautifully designed vintage pieces by small, even defunct, brands because they can be acquired for a fraction of the price of the most sought-after pieces by major brands.

“I try to share my knowledge,” he said, “what little I know.”