It's a difficult one OP because they really are allowed everywhere these days. My view is that if you have a dog, it's your choice and you run your life accordingly, not expect everyone to make allowances for you. But everyone seems to think it's the other way round - see the suggestions of getting therapy. I don't want therapy, I just want people who have a pet to be responsible and keep a close eye on them. I am not really scared of them anyway, I just don't want them around me.
Cafes are difficult. Although dog-free cafes do exist - or maybe allow them outside and not inside, that sort of thing.
As a pp has said National Trust places often have dog-free areas but you need to do your research. One example is Claremont, but that's at Esher so quite a distance from you. Not sure about the ones around the Bristol area.
Restaurants are often easier but if you just want tea and cake it's not necessarily a solution.
Don't go to garden centres. I can't think why a dog would find one interesting but people take them because they can.
Swimming pools are one option other than the Lidos on their last day but that's not an issue this time of year.
Agree a no dogs directory would be really good but probably change all the time as people back down on hassle from dog owners and allow them in. Occasionally it goes the other way when a dog poos somewhere and so the owner decides they are not letting them in anymore. My mum said a local cafe has just decided to stop them coming in again for that reason.
Wildlife areas seem like a good suggestion but even then they usually allow "well behaved dogs on short leads" which is fine until someone lets their non-well behaved dog off-lead.