As long as your account is securely linked to Apple Sign-In, Facebook, or an email address, you’ll be able to resume progress on your account from any device.
Apple Sign-In users will have to link their account to another source as well if they wish to continue play on non-Apple devices.
To do this, select your profile on the top left corner of the screen while viewing the cosmos and a window with options to further link your account will appear.
How to log in:
When you've downloaded the game to your new device, tap the profile icon at the bottom left of the Start screen (do NOT tap the Start button). Once you tap the profile icon, you will be able to log in to your previous account!