Bought and tried to install Shadowlands, but cannot find an INSTALL button
To play Blizzard games you need to Download and use the Blizzard Desktop Application (Launcher).
Article and Download Link here
Click the button for the game you want at the top of the app - in this case WoW. That switches to your WoW tab. On the left you will see the Install (if you still need to), or the Play button.
If you already had WoW on your PC then there is nothing more to download. All buying SL does is unlock the content that is already in the main WoW package.
If you are having issues with your purchase check your Transaction History in Account Management to make sure it went through.