I sat there watching this, frantically
Except that then a strange thing happened. The vehicle behind me started honking like crazy. I looked around and didn't see anything I was doing wrong, or any way that I needed to get out of the way to help the ambulance, which had plenty of room to get around me once the children were done crossing. I looked behind me, and saw that it was a school bus honking, and the driver was waving and pointing at the ambulance, trying to get the attention of the driver who was letting children out. So maybe the other bus driver did think the proper procedure was to somehow get the children out of the road.
It was an interesting little coincidence to come across, except that now I still don't know for sure which one has the right of way. At least I do know with certainty that had a mail truck also appeared, it wouldn't have even been in the running.