So i bought the cold war Cross gen UPGRADE on my Series S for £4.99 BUT...

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

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

Looks like you can upgrade to the cross-gen bundle:

https://www.microsoft.com/en-gb/p/call-of-duty-black-ops-cold-war-cross-gen-bundle-upgrade/9mxpqc207hnm?SilentAuth=1&wa=wsignin1.0&activetab=pivotverviewtab

Once you have this you'll need to go to the Mircosoft Store on your Xbox and re-download the Series X|S version.

I currently have the Xbox One S and the standard edition of Cold War installed. In a few weeks I’m going to get the Xbox Series S and I was wondering if I have to buy the Next gen version for Cold War at it’s full price.

Thank you

No you don't just buy the cross gen bundle for 10$