Error description: "Something went wrong with a file."
Error code: BLZBNTAGT00000840

A workaround that succeeded for me: Use GE-Proton9-23 or newer.

Details:

• GE-Proton can be either the one obtained from GloriousEggroll (which is made for use within Steam), or the one obtained from the Bottles project (which is more like regular Wine in that it does not use Steam's special environment setup script or have extras like DXVK built-in). Bottles runners are described here.
• It may require setting the Wine prefix's Windows version to Windows 11; I have not been able to confirm yet. This can be done on the winecfg Applications tab, or by running winetricks win11. Note that it requires a fairly recent version of Wine (and winetricks if using that method) so it might require temporarily upgrading Wine/Proton in order to choose that option. Battle.net might lose your saved password if you do this, in which case you would have to log in again.
• Once the games are updated, returning to the previous Wine version should be safe, at least until a better solution is found.

Alternative approach:

Mert Eren on the Codeweavers forum relayed another workaround. Summary: The problem apparently lies in the latest version of the Battle.net "Agent" program. If your installation is old enough to still have the previous Agent version installed, it is possible to remove the new one and block it from being reinstalled, relying instead on the old version to update your games and make them playable.

CodeWeavers are investigating for the sake of their customers, but have not yet announced a fix.

Ongoing discussions:

https://us.forums.blizzard.com/en/wow/t/couldnt-update-after-111-now-cant-install/2044960

https://old.reddit.com/r/linux_gaming/comments/1i1on1i/unable_to_update_world_of_warcraft_retail_error/