Plenty of trails on the Mendips/Rowberrow. These don’t drain well, though, so personally I don’t ride there much in the winter. If you don’t mind a 40 minute drive, Wind Hill is excellent, and the trails (especially the blacks) drain really well, so are rideable even during the wettest months.
Of the stuff in Bristol, Ashton Court is all weather, but very tame unless you’re on an xc bike. Leigh Woods is similar, although there is some off piste stuff which is much more technical. Gets very muddy in the winter, though.
Bike Park Wales is only an hour away and now the bridge is free of charge is a bit cheaper overall.
There are a couple of decent pump tracks in Bristol if that’s your thing as well.
JP