Here’s the FAQ page with the various possibilities of purchase transfers:
Please note that if you buy the game with your CoG account then you just have to tap the “restore purchase” button in the various app.
If you bought the game on Steam you can restore your purchases directly from your CoG account (under your games’ list there’s a link “restore purchases from Steam”).
If you’re encountering difficulties you can contact the support and they’ll quickly restore your purchases (you just need to send them the receipts).
@AnitaMurphy, kindly note that in-game progress is stored locally, both if you play on the app or on the website, so unfortunately you can’t sync it and resume your playthrough from one to another.
At the moment the only way to sync progress between different devices is to play on Steam.