I use "uni" or "university", no "the"! When talking to people about uni's where there are multiple Universities in question I refer to them by their place/location/other part of their title, such as Staffordshire, Derby, Nottingham, Birmingham (Uni)...etc etc.
I've never heard anyone here say "the University", and its usually spoken as a place, or an institution.
I guess it could be because that our Universities here have their location or other part of the name first, such as "Staffordshire University" and not "University of Staffordshire". I know we have some uni's which do it that way in the UK, but its mostly "University" being the last part of the name, therefore needs no "the"!
Hope this makes sense!