Leeds Castle in Kent (first mention in 857 AD) will be interesting for people who love historic buildings and Royal family stories. For example, future queen Elizabeth I spent some time prisoned there. Also a huge park with peacocks and other birds stretches nearby.
Milton Keynes is a good example of the “new city”. It was conducted only in 1967 during the 3rd rebuilding wave. There is no “old town” centre, city is divided into the squares with particular purpose for each: business, shopping, leisure. No traffic lights, only round crossroads. All parking machines work using the sun batteries. Also they have the longest shopping centre in the world. Is it interesting for you to take a look at the city of the new era?
Birmingham is the most populous British city outside London, but it is surprisingly “usual”. Major companies, universities and institutions are located here, maybe it is comfortable for living, but for tourist it is a city for one day visit.