UAfan with whom are you travelling onwards from T2? UA won't be able to help you at all if you're late, but if you're onwards with Star carriers it shouldn't be an issue.

Your experiences so far have been at extremes. In context, to be an hour late on a long haul with a non-protected onward ticket is pretty unlucky. To breeze through the connection process in 20 minutes is pretty fortunate. A 1hr 20 connection is a solid guess for a good result.