Only been to Norwich once but know a lot about Bath.
Bath has a fantastic quality of life, and is full of ex-Londoners who moved to get a better life. It comes at a cost though. Housing is limited, the crisis hasn't really hit Bath or the property market and prices are leaping 30% + in a year in some parts.
Is it Liberal? Kind of - large educated professional population, people in 20s and 30s are quite liberal and cosmopolitan. It's also sort of conservative and there's a lot of money around. It's generally very white and middle/upper middle class though.
Culture - definitely. Good to excellent restaurants (although fewer ethnic restaurants); excellent theatre and children's theatre does pre-west end shows as similar demographic; good independent shops; lots of sporting activities. Not so good for big name music of any sort or clubbing. Bristol is 15 mins by train for such things, and London 1.5 hours by train, although the last trains are remarkably early.
Meeting new friends - yes if you're a student, yes if you've got children (in which case it feels like a village as you know everyone). If you're single and child-free and in search of a partner and friends it's not great.
Traffic and parking are a major headache but you can walk most places as Bath's so small.
Happy to answer specific questions but your first step should probably be to look at property prices, Londoners are often in for a nasty surprise if they're hoping for a much cheaper life.