What are the obvious universities? Oxford, Cambridge? Durham is the third oldest after these two and comes highly placed in any league table of the UK. Lancaster, established in the 1960s, Bolton established in the 90s, like many other former colleges granted university status around that time.
If they come up with the goods, they stand or fall on their own merits. As costs rise, they are chasing less and less potential students who could afford it.
When it comes to cities, then it becomes a mystery to me. Does adding the title "City" to a post industrial town that once thrived in the age of Britain's manufacturing heyday magically turn it from a dreary place full of betting shops and discount stores with an ageing population, into a magnet for thrusting, youthtul investors? There are many such towns in Britain today, each with their own history and civic pride. How is the decision made? Who chooses? Whatever the outcome, there are many disappointed candidates. For football fans, if their team is named after the town, it gets to put "City" in its logo.
Nope. Even after a bit of thought, it's still a mystery.