I run two nights in Bristol,
A Man Walks into a Bar at the Lazy Dog in Bristol, it's a place were you can practice 5, 10 and 20 minutes sets as well as compere. £2 for 2 hours of stand up. https://www.facebook.com/bristolpunchline if you are interested in booking slots then email firstname.lastname@example.org
Comedy Combo at the Cube Cinema in Bristol https://www.facebook.com/ComedyCombo sets are around 5-7 minutes, It's 90 minutes of stand up and then a comedy film. They have shown the Blues Brothers, Dead Men Don't Wear Plaid and stuff like that. Email email@example.com to book a slot. Looks for acts a little more polished then A Man Walks into a Bar but has a big stage, mic and spot light and an crowd of between 70-100 people.
You should also check out Giggle Byte https://www.facebook.com/gigglebytebristol?fref=ts they aim to promot stand up in and around Bristol and post stuff about up coming gigs.
Hopes this is helpful,