The University of London is huge. The University/university has 1000s of students.
Is it correct to leave out the capital u in the second sentence?
The University of London itself seems to think it is required. Its website has this final paragraph on its About Us page:
As well as offering a world class education the University has a range
of other services on offer including a careers service, data archiving
and networking, library services, conference facilities and you can
even use University buildings for location filming.
The justification for this is that it is the University of London, and not any old university, that is being referred to.