Hey y'all-- I'm also waitlisted at UT Austin and managed to get in touch with someone at the department in early March (before Coronavirus, so this information may have changed). This is what I was told:
When someone we’ve admitted declines our offer, we accept someone from the waitlist. However, I cannot begin that process until 2 or 3 applicants turn us down. I have not yet admitted anyone from the waitlist. Whom we then admit depends upon the fields of those who decline. However, people have until April 15 to accept our offer. As a result, in the past we were not able to go to the waitlist until the end of March or early April. That pattern seems likely this year as well.
Not really any surprises with that info, but I just thought I'd pass that on! Best of luck!