Teens gain the ability to apply to university partway through the teen life stage, and all young adults, adults and elders should be able to. It doesn't depend on anything to do with graduation.
Is it possible that through your attempts you did get into the university but didn't realize it and haven't enrolled? Do you have the option to enroll in a university if you click on the mailbox or computer?
What happens if you take another young adult, adult or elder (just grab a premade or random townie from Manage Households and play them for a bit to test) and try to have them apply to university?
Any mods in your game?
@crinrict yes, if they already applied and were accepted (without realizing it) then I'm not sure they would see the apply option any longer. If that's the case they would see the enroll option.
