Until Local Government reform in the 1970s separated many cities from their historic counties, Northumberland had a city, and Warwickshire had two.
There are Roman Catholic cathedrals in several places without Anglican cathedrals: notably the cities of Lancaster, Leeds, Nottingham, Plymouth, Salford, and Westminster, and the non-cities of Aldershot, Arundel, Brentwood, Northampton, Middlesbrough, and Shrewsbury.
Southwark is not a city, but has two cathedrals.