Basically, as we all know now ever since the recent Steam update, matchmaking has been broken, and most if not all players haven’t been able to play the game. As far as I know, there are 2 solutions.
If you own the game on battle.net this issue is not present on it, therefore letting you play still like normal or purchase the game again for 30-60$ on the platform and be able to play again.
If you don’t own the game on battle.net and or do not want to repurchase it then you can try the solution a Steam member found
Step 1: Press Win+R and type in the box “steam://open/console” which should open a new tab called console on the Steam app.
Step 2: Once there paste “download_depot 1985810 1985811 5380296931597271327” which should start a download of around 40 GB of the game files before this broken update it won’t show any progress on the download but it will let you know once it finishes. It’ll download the files for this in the folder “C:\Program Files (x86)\Steam\steamapps\content\app_1985810”
Step 3: Once the download finishes copy the files that were just downloaded from C:\Program Files (x86)\Steam\steamapps\content\app_1985810\depot_1985811 to
C:\Program Files (x86)\Steam\steamapps\common\Call of Duty Black Ops Cold War or wherever Cold War is installed to on your PC. Once a prompt pops up asking to replace or delete the files replace them as this will be fixing the game.
Step 4: Once that is done go back to Steam right click on Cod black ops cold war, then Properties then update, and click the option to update only when starting the game.