No, it's the same everywhere. If one person is exempted in a two person household, that's a 25% reduction (the same as if a single person was living somewhere). If a person who lives alone is excempted, that puts the reduction down to 0% as technically no one eligible for tax is living there now. And yes, claiming something like AA is a prerequisite. Some people get lovely refunds due to some serious backdating.
Contact your water board about their reduction schemes. We didn't qualify for one as not on the right benefits and apparently we don't use up enough water for another scheme so wouldn't have saved but it's always worth enquiring if your water use has increased due to a lot of extra laundry you have to do.