Fastpitch123,
Yuck! That is an ugly situation for two umpires to get themselves into!
No matter what they decide, 1/2 the fans and players are going to be pretty ticked off.
As for who is right and who is wrong, the plate umpire is obviously wrong if he called the batter out when she was allowed to advance. (I'm assuming she was allowed to- you didn't mention the number of outs or if first base was occupied. At face value, the information in your post doesn't tell us if she was allowed to run or not!)
Some things to consider to sort this out:
One factor to consider: the defense is obligated to know the rules and to know if the batter is allowed to run or not. If she was, then the defense should follow through with the play and make the throw to first.
Advantage: offense.
If the pitch bounced and went right into the mitt, and the catcher was holding it right behind the plate, the batter most likely would have been put out anyway. The umpires might stick with the out call.
Advantage: defense.
The tie-breaker might go to whichever coach has the smallest meltdown when the call doesn't go his way!
Seriously, this is a perfect example of why plate umpires are trained to NOT say, "Batter's out" immediately following a third strike. Most of the time, the batter IS out. But if you get in a habit of saying this after every strike out, it's going to bite you on that one play- like this one- where the batter IS NOT out!
The only time an umpire should be yelling, "Batter's out", is when the batter is advancing up the first base line on a play where she IS NOT allowed to advance. Before uttering that phrase, the umpire needs to be 100% the batter is indeed out. This is a good game management technique that can put a stop to a lot of confusion before it even happens.