Leeds hasn't got a cathedral but the population grew.
"The difference between towns and cities
The difference between towns and cities is differently understood in different parts of the English speaking world. There is no one standard international definition of a city: the term may be used either for a town possessing city status; for an urban locality exceeding an arbitrary population size; for a town dominating other towns with particular regional economic or administrative significance. Although city can refer to an agglomeration including suburban and satellite areas, the term is not usually applied to a conurbation (cluster) of distinct urban places, nor for a wider metropolitan area including more than one city, each acting as a focus for parts of the area."
That's completely irrelevant - you've quoted a definition of the word "city", not something pertaining to City status in this country, which is something altogether different.
As the Department of Constitutional Affairs state:
"City status is not, and never has been, a right which can be claimed by a town fulfilling certain conditions. The use of specific criteria could lead to a town claiming city status as of right, which in turn might devalue the honour. All applications are considered on their individual merits."