This particular forum is for issues installing Classic Blizzard games such as Diablo 2, Warcraft 2, etc. So, no, they won’t address the issue here.
You seem to need assistance with a modern game made by Activision. If you have issues launching the game using the Desktop Launcher Application, which Blizzard makes, you can post on the Desktop App Tech Support forum.
If you have issues with the Launcher recognizing a game key, you can try the steps here Free Beta Cold War - #12 by Kaldraydis
As said there,
While we can of course still help out on this forum, troubleshooting for the Battle.net desktop app is best provided through the Desktop App Tech Support forum. This classic games forum is more for our oldschool games like Diablo 2 and similar.
If you have Technical issues once CoD:CW is launched, you need to contact the game maker - Activision.