I went to the Edinburgh Fringe festival last year and saw Marcus Brigstocke live, and he absolutely killed it. Probably the best act I've ever seen.
Aside from him, I've got to agree with Bill Bailey. I honestly think that he is almost genius-level in his musical ability - it's not just that he can play such a wide variety of instruments, but the way that he understands the theory of music so well that he can turn just about any piece of music into a joke, and any joke into a piece of music, is just astounding. It takes a very gifted comedian to write scripts as funny as his, but to write them while blending it all in with musical skits which are just as funny is an incredible talent.
Sorry to all the non-British people here, as I'm not sure either of the above have had any exposure outside of the UK. However, if you haven't heard of them then I thoroughly encourage you to find them on Youtube and see what you think.