This is exactly what happened to me just over a week ago from IAD to LAX (789). F was full three days before the flight and remained full during boarding and all F passengers had checked in.

While seated with my daughter and son (we were all on the upgrade list as the top 3), I noticed one F seat became available at T-3. I said to my daughter that I might get upgraded. 2 min later, the gate agent came on board told me I got upgraded but it was too late for him to print a new boarding pass. He asked me to move to a specific seat in F and told FA on his way out before the door was closed. While the plane taxied to the runway for take-off, the app showed I was upgraded and reflected the new seat number.