Northwestern city (9)
(Other definitions for lancaster that I've seen before include "Historical English royal house with the emblem of a red rose" , "Say Crouchback" , "University" , "City of the red rose" , "Ancestral (anag) -- English royal house" .)
(Other definitions for liverpool that I've seen before include "Where Beatles started" , "Merseyside port" , "Merseyside city of The Beatles" , "city with two cathedrals" , "Cathedral city" .)