You need to make sure that "Cloud Save" was enabled on the old phone and you are logged in with your Google Play account. If so, then on the new phone it is enough to log in to the same Google Play account.

If “Cloud Save” were enabled you can just install the game on the new device and sign in with the same Google account. If not, you need to enable it in the settings and complete one more level to sync progress.

If it still doesn't work, then it's the Google Play Games app (on the old or new device, probably both). You need to reinstall it and sign in with your Google account. After that, reinstall the game on the new device and launch it.