To establish a timeline for Rolex, it's important to clarify what you mean by "luxury brand." Rolex had several ground-breaking innovations as a watchmaker, including the the world's first waterproof wristwatch, the Oyster, in 1926 (U.S. Patent 1576120.) and world's first automatic date function, the DateJust, in 1945.

Beyond technical innovation, Rolex was certainly considered a "luxury brand" by 1954 when Ian Fleming gave James Bond a Rolex to wear in Live and Let Die ("Behind it there was a broken rock leading steeply upwards. It was the keel of the Secatur and Bond's heart thumped into his chest. He looked at the Rolex watch on his wrist."). For comparison, Fleming had Bond own a Bentley and drink Bollinger champagne, which certainly were luxury brands in 1954. Sean Connery wore the Rolex Submariner in Dr. No, giving iconic status to the dive watch style.

The innovative aspect of Rolex’s marketing at the time was to position its products as an “accessible luxury” – that marketing managers started referring to as the “new luxury”. This strategic approach has been very popular in the fashion and luxury sectors since the 1980s.

In the 1980s Rolex began a massive awareness campaign to subtly incorporate the Rolex name in everything related to time and success. Travel to any airport worth going to today and you can be sure you’ll find a lot of Rolex-branded clocks. The same goes for popular sporting and cultural events.

Whomever this quote is attributed to has, I think, confused 'Luxury Brand' with 'Statement of Wealth'.

Rolex, since the 1920's, has been a brand synonymous with quality. They brought out the first waterproof and dustproof watches, had the first automatic self-winding movements, and were the first to pioneer the idea of the datejust.

Sir Edmund Hillary and Tenzing Norgay both wore Oyster Perpetuals when they climbed Mount Everest in 1953, which led to the new Explorer range the following year.

In short, Rolex has always been associated with quality, and so their watches were always very expensive, but it wasn't until the 1980's that having a Rolex became a must for wealthy people.

Much like Porsche, Rolex really capitalized on the Yuppie surge of the 1980's. The image of the wealthy businessman, with his red Porsche 911, Motorola DynaTAC in one hand, Rolex on the other wrist became something to aspire to.

you should also try asking /r/watches.

There is another thread here in this sub that explains how the perception of Rolex has shifted. If Che Guevara was such a staunch opponent of capitalism, why did he often like to show off his Rolex?

In case you are not familiar with it, all Rolex watches that I am aware of, and close to all other brands luxury watches are mechanical. Before 1970's all commercial watches were mechanical. In 1969 Seiko produced the world's first commercial quartz wristwatch. They practically destroyed the mechanical watch industry. Most brands either adopted the quartz technology, went bankrupt or changed to luxury brand, as they could sell just afew. In later years most bankrupt brands were revived as luxury brands. Nowadays mechanical watches can be seen more as an art form or a statement than practical devices (Not that they are not practical, but the functionality, cost, accuracy etc. make them seem so ).