According to an interview with British GQ, the Driver is sporting a Patek Phillipe watch with a brown leather band, lumed hour and minute hands and sub seconds. It was a prop, custom made for the movie, so it’s not a watch you can go out and buy (not that many of us could afford it if it were real — prices start at $15 000).
Inspired by the movie, I was able to solve a problem I’ve been having finding the right strap for my 1940s Zenith. I added a brown leather band and voilà — I’m cool like baby goose. I just have to get a cool pair of driving gloves.