If this was a failing corner store I'd be worried--but I doubt there is a substantial risk that college won't get it straight sooner or later. Now as to major inconvenience to your family--I don't know how to solve that.
If wife is a lower pay nonadministrative worker, NY has a form/process for state dept of labor to step in and help as to nonpayments but I suspect that takes forever once you file it.
I can understand messing up one payday cycle at start, I'd even bite my lip at two--but 3 would take the cake.
Is wife sure she was added to system, signed up for pay, benefits, withholding etc--be sure and keep records.---
Practical issue may be that many such institutions put new staff on "90 day probation " or some such words and if you make a royal pain of yourself in that window you stand to get exited........
I once got hired and my scatterbrained boss fromHarvard forgot to add me to payroll and was trying to cover up his goof.
Another time in a more senior role a new monthly pay slot just happened to work such that I needed to wait some 7 weeks for my first check as one got paid at end of 1st FULL month and I started about the 4th. .