I bought the new BO Cold War Cross-Gen on the Xbox, and want to build a PC Later.

As Farscape has said all you need to do is link your accounts together.

I have COD Cold War on the new Xbox and Modern Warfare on my PC. My credentials are shared with both platforms.

whats that again? Cridentials?

just use the same cod credentials as you do on the xbox and you should be good to go.

From what I can tell, to access the Series X version of the game, you need the Cross-Gen bundle...once you buy that, you should then be able to download and install the Series X version. Without the cross-gen bundle it will, presumably, just install the XB1 version

From the Store page:

"For digital owners of the Call of Duty®: Black Ops Cold War Standard Edition, this upgrade will unlock the Call of Duty®: Black Ops Cold War Cross-Gen Bundle, giving access to the Xbox Series X|S version of the game. After upgrade purchase, search in the Xbox store for the Call of Duty®: Black Ops Cold War - Cross-Gen Bundle to install the Xbox Series X|S version of the game."

Hi,

You have to pick up from the store the cross-gen edition and it will cost you the difference between the standard and the cross-gen version (around 10 bucks) :

How to upgrade Cold War from Xbox One to Xbox Series X/S