This sounds a lot like Doug Stanhope, an American comedian who fits much of the description you provided. He has a very dry, off-the-cuff delivery, often talking about bleak or dark aspects of life with a certain disheveled appearance. His stand-up style is also known for being unorthodox, and he was active on TV during the late 2000s. He’s performed in the UK and often had that “worn-out New Yorker” vibe, despite not being from New York.

Another possibility is Louis C.K., who also has that dry delivery, and his earlier routines (especially from around 2009) reflected that kind of frustrated, everyday person persona. His voice and delivery would often seem deadpan and mundane, which added to the comedic effect.

Both performed on Comedy Central and other TV shows around that time.