First, some caveats about my experience:
- it was solely as a user, not an operator or administrator;
- the systems are smaller than what you are asking about (~20 workstations at high school, maybe 100 in university);
- it dates towards the tail end of the era you are asking about (I graduated high school in 1997 and started university in 1999).
At my high school, we had a computer lab with about 20 PCs. The computer lab was originally set up for running DOS, which was kind of the standard for the time – if a teacher had any computer experience at all, it would have been with DOS or Windows; Unix was not a thing in high schools or general society, which is after all where teachers come from.
However, one of our teachers was a bit of a unicorn: he had been a maths professor and researcher at university, but he grew disillusioned with the quality of knowledge of the students. So, he decided to become a high school teacher instead to have an impact earlier in a student's education.
As a former university maths professor and researcher, he was intimately familiar with Unix, TCP/IP networking, and system administration, and he converted the whole lab to Linux. There was a slightly more powerful PC for the teacher which was running Linux, and all the student PCs would netboot either Linux or DOS off of the teacher's PC (there was no separate server, the teacher's workstation also acted as fileserver, DHCP, DNS, etc.), depending on the teacher's preference and/or the subject of the course. All filesystems were network-mounted, even in DOS. However, there was no persistent file storage for students, instead we carried around one FAT-formatted and one ext-formatted 3.5" floppy disk. OTOH, assignments were handed in by copying them onto a network share.
At university, the setup was much larger, but similar: there was a mix of RS/6000 and HP-UX workstations, and Linux PCs. Everything was netbooted and net-mounted.
So, in both cases, only one copy of software had to be updated. In case of the high school lab, it was only turned on during classes and there was no remote access, so scheduling updates was easy: walk into the room at the beginning of a period, if nobody's there, you have a 45 minute maintenance window.