The Guardian has a short video (can't embed) showing martinis being made at 3 London bars; Mark's bar at Hix, the bar at Dukes Hotel and the bar at the Connaught.
The standout for me is the Dukes barman Alessandro Palazzi with the cocktail cart. Duke's bar was apprently a hangout for Ian Fleming (even where he came up with the line "shaken not stirred", though the iced fixings don't need shaking or stirring) but forget all that - Palazzi pinches the life, and oil, out of that lemon. Great stuff. Watch.