If you’ve checked that the purchase went through successfully in your Transaction History, and you’re not seeing the option to Install the game, this should help:
- Log out of the Battle.net App in the menu on the top-left
- When given the option to log in, click the Cogwheel above where you enter your e-mail
- Click ‘Continue without logging in’
- You should be able to install Warcraft III at this point
Once it’s installed, you can log back into the App and it’ll update that you have the game ready to go.