Start indoors, gradually getting your cat accustomed to the harness and leash. Move to an enclosed outdoor space (if you have one) as this will be a potential new experience for your cat.
A harness and leash are essential, as you cannot predict everything and safety must come first for your cat.
It's imperative that you remember that your cat may not like the harness, and that's okay. Never force a cat that's unwilling to comply.
British Shorthairs are similar to American Shorthairs, but less open to walks. While you might have success, this breed prefers to stay at home.