The Royal Charter is the key to citydom. There are towns with cathedrals which are not cities (Bury St Edmunds), and cities without cathedrals (Sunderland).
I live in Swindon,which has a population of about 180,000,so i`d say it`s a largish town. We have neither a cathedral,or a university,but several times we`ve applied for "city status" via Royal Charter. Lost out each time,most recently to Sunderland i think