The launcher is actually an Internet Explorer webpage, so even if your default browser is something else, it's actually IE you need to fix.

So from the Internet Explorer options, from the first "General" tab you want to delete the browsing history, specifically "Temporary Internet files and website files" and "Cookies and website data". Also untick the box for "Preserve Favourites website data" to make sure it cleans up completely.

After doing this try loading the game again. If it's still not working go back into the options and on the advanced tab and there should be a "Reset" button at the bottom. Clicking that should force things back to working.