But not “chapel”.
I live in Chapel Hill, am an alum, have college age kids, know kids that currently go to UNC, spouse works at UNC, and am here to tell the OP that no one says just “chapel” when referring to UNC. Most people say UNC or Carolina. Sportscasters, of course, say North Carolina. Some people being thorough say UNC-Chapel Hill (but we all know that’s the real UNC and all the other UNC’s get an extra letter ala UNC-C, UNC-W, UNC-G). I don’t mind hearing “Chapel Hill” when referring to the school, but it does sound very old school to my ears. Native NC-er here.