Westminster is the base for the Archbishop of Westminster who is the Head of the Roman Catholic church in England and Wales. This is currently Cardinal Vincent Nicholls - the former Archbishop of Birmingham. The Head of the RC church in England and Wales is often named a cardinal after appointment as Archbishop of Westminster.
There are other cathedrals in London because of the size of the City but I wasn't going to cloud the issue
Cardinal St John Henry Newman is a different story. He was not based at St Chad's. He started as an Anglican clergyman, converted to Catholicism in 1845 (I had to look that up), became a Catholic priest and eventually founded the Oratory in Birmingham. He was made a Cardinal in recognition of his work for the church in England. He lived at the Oratory until his death.
Hope that answers some of your questions. We are now off topic so I will stop there.