The difference would be trivial since the mail would still be going through and sitting on Google's servers. Client has to fetch it from some server and your university using Gmail means they've already outsourced the whole email system to Google.

I'd take u/anticonnor's advice and gradually move services and correspondence to your new mail provider. Several years back, Google flooded the education market with cheap cloud services and "unlimited" storage. Then a couple years ago, they started charging huge premiums on storage use above a certain limit, leading to mass data deletions and the discontinuation of alumni email among many universities. Who knows when they'll pull the rug again.