Endless loop of iCloud prompt rendering iPad unusable
I found a (much less than satisfactory) solution to the problem.  There is a slight delay between when pressing "cancel" on the iCloud password prompt and when the next popup opens.  I was able to navigate to the WiFi settings, and after a lot of frustration, was able to connect to my new wireless network.  After doing so, the prompt accepted my updated password.  Quite annoying.