We Americans do use the term "undergrad" to mean someone who is working toward a bachelor's degree, as opposed to a "grad student" who already has a BA and is working on a master's degree or doctorate. But those terms are heard only at institutions that have both. Otherwise they'd just be called "students."

And I THINK it used to be a rule that an institution was called a "college" if it offered only a BA and a "university" if it offered higher degrees. But that distinction has become blurred in recent years. Many institutions have upgraded themselves from "technical colleges" [which used to offer only 2-year "associate" degrees] to "colleges" and from "colleges" to "universities." I have no idea how they justify this, or if they bother to justify it at all.

Maybe this looks like thread creep, but it is about "pomp and circumstance" after all!