In universities, computers are not an essential service
This is going to sound very odd, but it really is true: in most universities, computers and networking are not a truly essential priority. I don't mean that computers aren't important or that losing computing would not be a very serious problem, because in a modern university neither is true; if the university as a whole or even a department were to drop off the network it would be a very big deal and a crisis.
But in the worst case, if the university's computers all went away one day the university would not shut down until it could replace them. There would be major disruption and pain, but people would keep on getting taught and a fair amount of research would still keep happening (probably more than I would expect). It definitely would not be the kind of event where you tell everyone to stay home until further notice because there is no point in them showing up to work.
Partly this is intrinsic in what the core mission of a university is. A university exists to teach undergraduates, get graduate students to produce theses, and to obtain grant funding; none of these functions universally require computers (although in some fields they do). Another part of this is due to the patterns of communication inside universities, where for many professors and graduate students it is more important to communicate with people outside the university than most people inside it.
(This leads to a situation where the disaster recovery plan for many people would be 'take my personal laptop to a coffee shop, get a webmail account, and start mailing people from it to tell them my new address'.)
The one exception to this is HR systems, and in particular payroll. If the university cannot pay people their salaries somehow, it will stop having very many people before too long; not necessarily because people want to leave, but more because there's only so long that people can go without being paid before they have to find another job to cover the bills.
(I'll admit that I'm somewhat handwaving the issue of essential data like course records. In the medium term a university without access to its computerized course records might have problems giving out undergraduate degrees, which would mean that its undergraduates would start evaporating.)