To add on a little bit more info, when I moved some characters and worlds from one device to another, I needed to have the iCloud Drive (Settings>iCloud>iCloud Drive) turned on for BOTH devices (in my case my iPhone and my iPad) BEFORE moving local data to the cloud data on the first device in order to see it on the second device. If I remember it correctly, I couldn't see the character or worlds within the game's cloud options unless I did that first. Did you happen to do it that way?
Also, it's a simple oversight, but make sure you are connected to the internet and logged in with the same iCloud account. You do need to be logged in as the same Apple user on both devices.
Once you uninstall the game, or delete local characters/worlds, the local files will be gone forever. So make sure that you have them on the new device before getting rid of them on the old one.