Hi,
Thank you for writing to Microsoft Community Forums.
We understand your concern as the PC that is used by your mother has changed from Windows 10 home to Windows 10 S after BIOS update and now it is not upgrading, because Windows 10 S is not activated, and the operating system became a non-core edition.
In order to change the Windows 10 from non-core to core edition, we would suggest you to perform these steps and later sign in with the same Microsoft account that you used before the BIOS update.
Put the PC in software clean boot, refer the article How to perform a clean boot in Windows. A ‘clean boot’ starts Windows with a minimal set of drivers and startup programs.
Note:After you finish troubleshooting, follow the sectionReset the computer to start normally after clean boot troubleshooting to reset the computer to start normally.
Once the above steps performed, then we would recommend you to download Windows 10 2004 version installation media (ISO) using the media creation tool on the desktop of the PC and then refer this article by PaulSey to run an in-place upgrade/repair.
Note: Please backup your important data that are saved in C drive to any other local drive or external drive before performing In-place upgrade.
Let us know the results. If the issue persists, we will certainly assist you further.
Aditya Roy
Microsoft Community – Moderator