The app does not work on Castaway?
I can share my experience on this, which MIGHT explain why some people find that the App works on CC and some find it doesn't.
When I first got off the ship, it worked fine. Then, when I was on the tram (and thus a bit farther from the ship and moving) it did not work. When I got to beach, and tried again (and was thus a little closer to the ship or perhaps on-land
DCL wifi), it worked fine again, and continued to work throughout the rest of the day.
So if you have trouble with it on CC, keep trying. It may work again in other parts of the island.
You do need to keep checking it to see if you have a message. Also, of the 9 people in my group, one or two had more trouble with messages getting through than the others did, so there is some variability that did not correlate to type of phone or anything like that.
For people following this thread with younger children, I will say that the younger kids clubs still use the wave phones, not the app to contact parents to pick up their kids, but since your kids are teens, like mine, that won't affect you.