Im late in paying my university tuition.
How will that affect my credit report/score
Even though I don't carry a credit card?

Should not effect it what so ever, has the college threatened to report this or turn it over to collections

Will not go in your credit report.

Your credit score is affected by items reported to the credit bureaus. If your university reports late tuition, then it will have an affect. If they don't, it won't.