Go to the game hub for the PS4 game. Select the upgrade offer on the game hub and review the upgrade offer. Select Download (or purchase with the discount price) to download/purchase content. After the download is complete, you can play the PS5 digital game by inserting the PS4 game disc.

Yes, Call of Duty: Black Ops Cold War 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.