Black Ops Cold War old gen to next gen

Yes they should be transferred , as long as they're linked to the account!

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."

https://www.microsoft.com/en-au/store/p/call-of-duty-black-ops-cold-war-cross-gen-bundle-u/9mxpqc207hnm/0010

This is the product that is the upgrade from Xbox one Cold War to series x version

From Activision:

"Yes, Call of Duty: Black Ops Cold War (Cross-Gen) is cross-play, cross-progression,
and cross-generation enabled. Once you upgrade from the current generation console
to the next generation console, your profile, progression, and stats will move with you."

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