Minecraft PE will not update from 0.12.1 to 0.12.3

When I attempt to update from Minecraft PE 0.12.1 to 0.12.3 in the iOS App Store, clicking the "Update" button briefly shows a progress indicator before turning back into an "Update" button.

This issue appeared after updating to iOS 9.1, but I'm not sure whether it was an issue before the update. I believe that auto-updates were working before this, though I'm not the primary user of this device or Minecraft PE.

I went through various possible fixes with Apple Support, including restarting the device and logging in and out of the App Store/iTunes account. Their suggestion was to try uninstalling and reinstalling the application, but I'm not sure whether that would lose my kids' worlds.

Is there a way to uninstall and reinstall Minecraft PE on iOS without losing the saved worlds? From what I can see, my kids don't have any sort of Mojang account configured, so I believe the worlds are only stored on the device itself (and in backups). I'm not sure whether Minecraft PE supports logging in to an account anyways.