Yes, you can recover your progress even after reinstalling!
When you redownload the game, it will ask you if you want to recover the previous progress. Simply tap "Yes" and your old account will be reloaded!

Alternatively, if you tap "No" by mistake, or if you don't see this message, you can recover a previously saved progress by logging in.
To save your progress, you must first login through Facebook, Google Play, or Game Center on the original device with the progress you want to save. Once your progress is safe, you can recover it from other devices by logging in with the same account!
Just open the game, finish the tutorial and login with the same account in the settings panel!

Keep in mind that if you are changing your device's platform, you must use a Facebook account. If you are changing from an iOS to an Android, or vice-versa, Facebook is the only way to transfer your account. That happens because Android devices do not support Game Center accounts, and iOS devices do not support Google Play accounts.