If you have created a Rovio Account before it was discontinued on May 2018 you can still use it to synchronise your progress between multiple devices. Just sign in to your Rovio Account on the new device to see the progress on your account.
If you have not created a Rovio Account, please submit a request to the player support from the link below. Include your Rovio ID for the old and new device and your progress can be manually copied over. You can see your Rovio ID on the loading screen of the game as well as in the top portion of the totem pole inside the Nest.