Hello Jon.
I don't know round there.
The Folk Music Map lists a few though some info might be two or three years old now:

Nailsea Folk Club Singers Nights
Folk Club / Regular Event

Chipping Sodbury Folk Night
Folk Club / Regular Event

Saltcellar Folk Club
Folk Club / Regular Event

Downend Folk Club
Folk Club / Regular Event

Nova Scotia Folk Club
Folk Club / Regular Event

Golden Guinea English/mixed session
Folk Club / Regular Event

Dragon Folk Club
Folk Club / Regular Event Map

find the links on the map
(open in a browser, not 'maps')
https://www.google.com/maps/d/viewer?mid=1d1L8gNDhc6JP84Wyt01-LSEcL_A&hl=en_US&ll=51.46855215268434%2C-2.5907093999999233&z=11

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Mr Red will probably be along soon with a better link but...
Here's a link to one of his pages as a starter.
Should be the clubs page:
http://mister.red/fcc.htm