Have you tried checking application status on a computer? Whenever I get tired of waiting for the acceptance letter, I just go through the computer. It's like sometimes it just gets stuck in the mail and takes forever, but on the computer there's generally a result available in just a day or two, and you can enroll from that result without waiting on the letter. One time my Sim's official acceptance letter was delivered just a day before her first term finished!
I don't have high school though, so there could also be a glitch specifically in the early graduates.
In my game teens can't go to University, they need to be young adults. They can apply while teens but have to age up to attend.
I don't have HSY so unless that changes the rules for teens going to uni I think you'll need to age up your Sim and then you should find they can start then.
