When you sign in with Google Play Games, your Kingdom Rush Vengeance progress will be saved automatically. This means that you'll be able to pick up where you left, even if you uninstall the game or switch phones, as long as you use the same account and have a connection to the cloud.
Before you start
Make sure you have Google Play Games installed and updated.
Here's a troubleshooting for you to try
Open your game, go to the Options Menu, and check that Google Play save is enabled.
Once it's on, go back to the Options Menu. Tap Achievements Select your profile image in the upper left.
Google Play Games will open and show you your current profile