It was not product placement as we think of it today, i.e. paid product placement; sometimes filmmakers use products that the character would use without any other motivation, in order to add depth to the character. (Aston Martin didn't pay them, either.)

In 1962, pretty much the only watch that someone in Bond's role would have worn was a Rolex -- the whole idea of a waterproof dive watch was pretty novel at the time, since Rolex invented the idea in the 50s, and was already associated with intrepid adventurous types after Hillary wore one on Everest (again, because pretty much nothing else was up to the task).

I believe the Bond wore it Dr No on a NATO (nylon) strap, not leather; it's definitely a NATO in some of the later films, but shifts to the iconic bracelet eventually (e.g., certainly by the time Roger Moore wore one with an absurd magnet and unbelievable spinning saw in Live and Let Die a decade later).

Anyway, that site isn't a great reference. It's got some things wrong. The traditional story about the watch in that film is that they got to the beach scene without a watch for Connery, or without a proper one, and the scene was shot with Cubby Broccoli's watch to get at least the right brand in the shot. That story has been printed and repeated a shitload of times, but all the principals are long dead so I have no idea if it's been verified properly in recent times. (In Googling for references for this reply, I saw a new version of the story, which is that the watch was Connery's own; I've never read that before.)