Prime Minister familiarly welcomes northern old city with French greeting (7)
I believe the answer is:
bonjour
'french greeting' is the definition.
'prime minister familiarly welcomes northern old city' is the wordplay.
'prime minister familiarly' becomes 'bojo' (informal name for Boris Johnson).
'welcomes' is an insertion indicator (in sense of 'invites in').
'northern' becomes 'n' (abbreviation e.g. N. Ireland).
'old city' becomes 'ur' (ancient city state).
'bojo' placed around 'n' is 'bonjo'.
'bonjo'+'ur'='BONJOUR'
'with' acts as a link.
(Other definitions for bonjour that I've seen before include "French version of Hello" , "Good day, monsieur" , "Good day (French)" .)