At Filton -> Bristol Aero Club and Aero's. BAC can only be joined if you're a member of BAe, RR or Airbus so unless you work for them, you can't join. Also, they only have 2 aircraft and run as a non-profit organisation.
Aero's is a bit more pricey but if you live near Filton, then I think they're probably a good bet.
Bristol Intl -> They have a few flying schools mentioned above but again they're slightly pricey. Lulsgate seems to be trying to price itself out of the GA market. Then again though, if you manage to find a decent price to do your training, then go for it.
Other nearby places include Gloucester and Kemble so check them out. I don't know what transport you have available for yourself so you will have to weigh up budget and travelling distance. Also, which part of Bristol are you living in?