Never Alone Ki Edition added iCloud support in its 1.03 update. If you are upgrading to a new phone, you'll want to update the game to this version before switching over to your new device.
Go to Appstore > Purchased. Click the cloud button to download the game to your device.
Log in to Game Center in Settings. It is in the list on the left side in the same group as Music, Videos, Phots & Camera.
Once all that is complete, launch Never Alone: Ki Edition. When you press continue, and enter a level, your save data is uploaded to the cloud.
Android / Google Play:
Ensure your old device has the newest version of Never Alone Ki Edition installed. Go to the Play Store, Search for Never Alone: Ki Edition. If the app has the 'update' option, press it. This will add cloud support.
Once that finishes, launch the game. When you press continue, and enter a level, your save data is uploaded to the cloud.
Download the game from the Google Play store on your new device. When you launch the game, and select continue, you will pick up from the same point as on your old device.