The Ur Example is, of course, John Constantine of Vertigo Comics. He first appeared in Swamp Thing and spun-off into his own series some years later. (Swamp Thing showed the title character and John Constantine both active in the Crisis on Infinite Earths, although Constantine had only a very small role in the actual miniseries.) Constantine was created by Alan Moore based on his idea of "creating a character who looked like Sting", although Sting did not tend to wear trenchcoats or smoke (at least in public).
Though many writers have wanted to use Constantine in their series, DC denied them up until 2011 for fear of disturbing the "realism" of Hellblazer, even though Constantine (technically) inhabits the DC Universe, where a number of these Expies also reside. So, instead, various analogues and homages to John have appeared.
Constance Johanssen was also Ellis' homage/parody of Constantine in the Pryde and Wisdom mini-series: "Constance Johanssen. Excellent occult detective. Has a habit of getting her friends killed. Two hundred at last count"
It's also implied that Constantines gravitate towards this trope especially if their initials are "JC".
With the breaking of the Vertigo embargo in 2011, John's reclaimed the Trenchcoat Brigade role in the main DCU.
Pete Wisdom from Excalibur (even when he wasn't being written by Warren Ellis, his creator) was always like this. He's cleaned up a bit recently, though. Given up smoking and everything. Some fans have theorised that the original character was the Nineties Anti-Hero version of the character, and this is just a natural progression.
Cal McDonald from the Criminal Macabre comics by Steven Niles isn't British, but otherwise fits this trope to a T.
Marv from Sin City is this trope in its most violent shade. (Of course, almost every protagonist wears a trenchcoat in Sin City. Frank Miller loves this trope.)
Samuel Burke from Spawn. Although overweight, the guy sees everything in the shades of grey. He smokes from time to time and when he taggles along with Spawn in his adventures, he becomes an Occult Detective.
Though he does subvert it a bit by the coat being white.
...as does Felix Castor from The Devil You Know and its sequels by Mike Carey (given the fact that Carey wrote Hellblazer for a while).
Cal McDonald (see comics examples above) has also appeared in a series of novels by creator Steven Niles.
Harry Dresden has definite shades of this. Considering John Constantine has been around since 1985,and Jim Butcher is a self-proclaimed fan of the series, the real question is whether it is a case of "influenced by", "inspired by", or Captain Ersatz.
Though given Harry's stupid sense of humor, he might be closer to a cross between John Constantine and Spider-Man. (Which he has also admitted: "I follow the Tao of Peter Parker," anyone?)
Given Harry's status as a Pop Cultured Badass, he was probably trying to invoke the image intentionally.
A recent addition to the brigade is Jimmy Stark, antihero of Richard Kadrey's Sandman Slim. Ragged & snarky, with a dark past and hellish powers; wearing a black silk trenchcoat with no irony, and smoking Maledictions, the cigarettes from Hell.
Fitz Kreiner, from the Eighth Doctor Adventures. Being Genre Savvy, he seems to know it, too. He smokes, wears a leather coat most of the time and a trench coat some of the time, swears more than basically anyone he knows, has Perma Stubble (because he's bad at shaving), and is a lower-middle-class Londoner and a GuileSidekick. He's also basically quite sweet and sensitive, but most characters, upon first meeting him, distrust him.
Castiel from Supernatural was influenced by Constantine's image. Though he's not British and doesn't smoke (as far as we know), he does wear a trenchcoat and is morally dubious, even though he's an angel.
Most accurate example; Alucard from Hellsing. The guy's a cold hearted bastard who loves to taunt his foes, but occasionally sides with the good. You might as well call him the John Constantine of manga.
Damien from Nightschool. He doesn't have stubble, but when you can stop bullets from reaching your skin, break bones by thinking about it, and can flatten NYC then you can be a nitpicker.
Detective Badd from Ace Attorney Investigations, whose ragged trenchcoat is pockmarked with bullet holes from all the times he's been shot at. He doesn't smoke, though—he just makes constantly sucking on lollipops look cool.
To elaborate a bit on this guy, EVERYONE, including Manfred von Karma, respects this guy, even if only grudgingly. He's one of only three known people Franziska won't ever whip. To the fan community, he's Ace Attorney's very own Chuck Norris.
Riff of Sluggy Freelance fits most of the trope. His dark and troubled past working as a Hereti Corp freelancer is often brought up even today. His inventions potentially destroying all (or most) of mankind is another for which he's visibly trying to redeem himself for. He also has a string of girlfriends he left behind. He doesn't smoke though.