There are a lot of cathedrals and abbeys in the UK. Many cities have them and there are all sorts and all styles. There are a few modern ones such as the one in Coventry, rebuilt after the old one was destroyed in World War II but most of them date back to between the 12th and 15th Centuries with a few in subsequent centuries like the 18th century St. Paul's in London.
Many Abbeys were destroyed during the Reformation in the 16th Century but a few have been rebuilt or were converted to the new religion such as Westminster Abbey and Bath Abbey.
I've been to quite a few, listed below and I try to make a point to see one if at all possible. The existence of a cathedral gives a town city status even if it's small, like Wells, but it's not a requirement for city status. There are towns with cathedrals that are not cities and vice versa. Cathedrals are usually the seat of a Bishop or Archbishop. There are a few Catholic cathedrals but not many, and most are Church of England/Ireland or the Church in Wales and Scottish ones are referred to as Episcopal Cathedrals. There were no CofE cathedrals built between 1542 and 1836 according to this website.
Here's a list of the ones i've been to:
Inverness, Exeter, Salford, Lichfield, Liverpool (C of E), Southwark (London)
Canterbury, Cardiff (Llandalf), Chester, Coventry, Edinburgh, Glasgow, St. Paul's (London), Westminster (London), Manchester, Salisbury, Wells, Worcester, York Minster.
I've also been to Bath Abbey and Westminster Abbey which are of cathedral proportions and not ruins.
I think so far, Glasgow is my favourite followed closely by Wells and Canterbury.