As Servant mentioned, exterior of building (at least original 13th century one) is public domain, I just wondered if it would be possible to use name of Manchester Cathedral with its original exterior but with completely different interior.

possible to use name of Manchester Cathedral

The name is a trademark of the Church, you'll need permission.

with its original exterior

Considering what already happened with that building, it is almost certainly legally protected.

Really, this needs a lawyer's care to get permission.