Call of Duty: Black Ops Cold War is one of those titles that when purchased on consoles Current or now Old-Gen, automatically include the free upgrade to Next-gen. However, it seems that many users have accidentally downloaded the PS4 version, which is already backwards compatible with Sony's new console PS5.
Apparently, a flaw within the PlayStation Store, would be giving priority to the installation of the Old-Gen version over the new version optimized for the new Sony flagship, with faster loading and able to reach 120 fps. Currently, the solution is: in fact, just go inside the store and start the installation of the PS5 version manually.
However, some reports from users tell of slowdowns or errors in the download, with the latter solvable through a drastic console reset.