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

I might have found a workaround:

• Set the Wine prefix's Windows version to Windows 11.
• 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.
• This seemed to be what solved it for me, but it's possible that a side effect of my experimenting was the actual fix. I don't have a convenient way to confirm it.

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/