Just leaving my solution here how I fixed this issue, in case anyone else needs it:
Step 1: Delete all files in …\Engine\Binaries\ThirdParty\ShaderConductor\Win64
(Obviously delete engine files at your own risk, I made a backup of them before trying this…)
Step 2: In the Epic Games launcher do a Verify for your engine version.
This downloaded the files for me again and then everything worked as expected.
Just doing a Verify in the first place without deleting the files first did literally nothing for me…
This issue was caused by a Windows 11 update for us.
I just wish Epic Games and Microsoft could work together somehow, so that Windows updates don’t regularly break Unreal Engine. I feel like I’m starting to get too old for having everything break down in the middle of big productions at regular intervals. 