[EDIT: It seems that, despite the official blog post urging players to "download Zombies Free Access today", the free trial is not yet available to pre-load. We'll be sure to keep you updated as soon as it's available on the PlayStation Store, Microsoft Store and Battle.net.]
Treyarch has confirmed via a Tweet (shown below) and an official blog post that a free week of Zombies mode will be open to all players on the Microsoft Store, PlayStation Store and PC via Battle.net from the 14th January until the 21st January.
The free week will give players access to two modes (three on PlayStation): Die Maschine, Cranked and Onslaught (on PlayStation platforms only). All progress earned during the trial will be stored and transferred over to the full release for players who purchase it afterwards.
Trusted independent Call of Duty news source ModernWarzone, from a Microsoft Store listing, leaked the upcoming free week, and it seems their assertions were correct. However, Treyarch has also been teasing a zombies-related surprise on "115 day" (15th January). It remains to be seen whether this is the surprise, although it's more than possible.
The news comes as the game is preparing to launch its mid-season content update. Detailed in this official Call of Duty: Black Ops Cold War blog post, the update will bring new maps and modes to both zombies and multiplayer.
The free week of access to the Zombies mode of Call of Duty: Black Ops Cold War will run from the 14th to the 21st January on the PlayStation Store, Microsoft Store and Battle.net.