For even further confusion "London" actually contains two cities: London and Westminster. London was a walled city but Westminster was not. So "London" was we know it today is more like Westminster than London.
A cathedral is neither necessary nor sufficient for city status. City status in the UK is given by the monarch and that's all there is to it. Cambridge is a city without a cathedral and Bury St Edmunds is a town with a cathedral.
That has a cathedral too.