There's a good live jazz bar in Tower Bridge Road on a Thursday evening - well there was when I last went, which was in the summer. I can't remember the name, the River Bar I think, it used to be the Copper. Its at the Tower Bridge Road end.
Not quite SE1, but the Three Stags pub in Lambeth Road (opposite the Imperial War Museum) has jazz and comedy nights. I think the jazz is on Thursday and the comedy Friday. There is also something on earlier in the week I believe.