If you go through any sort of formal umpire training, you will be told that on a called third strike to only say, "Strike", or "Strike three" and NEVER say, "Out", or "Batter's out" in conjunction with the call.
The reason this is taught is illustrated by your play. If you routinely say, "Out" as part of the call, you run the risk of calling the batter out when the ball isn't caught, she isn't out and can advance to first base.
You probably didn't have an experienced or well-trained umpire behind the plate if he's calling third strikes like that. And, you probably didn't have an experienced crew working the games either, if the base umpire "overruled" the plate umpire!
The base umpire can't do that. This is the plate umpire's call and no umpire can EVER overturn another umpire's call. One umpire can consult with the other umpire to see if he saw something he didn't, then the umpire who made the original call can change it.
These are small points, but points that can really screw up a game if the umpires don't follow them!
Since the "reversed" call has put one team or the other at a disadvantge, the plate umpire is obligated to correct it. He has to ask himself which team was disadvantaged and try to come up with a remedy as close to "what should have happened" as he can.
On this play, since the ball was in the catcher's mitt and the catcher reacted to the bad "out" call, you should probably enforce the out. In all likelihood, the catcher had ample time to make the throw and beat the runner.
There might be a different "fix" if, for instance, this was a passed ball or wild pitch with the catcher giving chase back to the screen. In that case, the runner would have likely been safe anyway and you could keep her at first base.
The easiest "fix" is to not call batter's out when they're not really out in the first place!