I can recommend:
Those are all on the fairly trad side. If your idea of folk music is American-style singer/songwriter/guitarist stuff, try Bunjies Folk Cellar if it's still going, can't help much otherwise on that type.
- Old Rose and Crown, Walthamstow - good strong club almost always guest nights except in the Summer.
- Islington - always guest nights, resident band, very assorted residents, gets a bit eccentric sometimes
- Sharps - full of strong unaccompanied traditional singers, very friendly and always lots of fun, almost always singers nights.
- Cellar upstairs - good small club
- Musical Traditions - great place to hear real trad singers you may have never heard of, run by Peta Webb and Ken Hall, hence Irish/American bias. Tiny room.
- Court Sessions, Tooting. Good if you're in South London.
- Herga - excellent reputation, I haven't been there for about 25 years, must do something about that....