As far as what files are in C:\Windows\System32 I have:

fireface.exe
fireface.inf
fireface_asio.dll
fireface_asio_64.dll
TotalMixFX.exe
TotalMixFX.chm

I'm running Win Pro 10 x64 build 1809

Thanks MC - when the computer re-started following the upgrade, I didn't see the icons in the notification area, and the red Connection light was illuminated on my Fireface. Fearing the worst, I tried to use the installer to re-install them, which failed. Dragging the files manually from the Windows.old folder's System32 folder to the "real" one seemed to fix things, and allowed me to uninstall and then re-install them, just to be sure. Reading your answer, I see now that all I needed to do was re-start once or twice. It seems that a lot of issues with Windows are solved by simply waiting, and then waiting some more, then a little more, finally re-starting (and re-starting, and re-starting...)

However, I now have a functioning Fireface 800 working under Win10 1809 which, due to compatibility issues with Pyramix, is where I'll be staying until the next update from Merging. This is on my "guinea pig" machine (found abandoned by the dumpster behind my apartment, which nevertheless is fully functional - some people!), where I'm testing out Win10 before actually upgrading my main work machine, which has its own issues being a Bootcamp install on a Mac Mini. When I finally get around to doing this system, I'll remember your advice, or simply uninstall the RME drivers prior to doing the upgrade, and then reinstall them once the dust settles.

Thanks everyone!

I've been wary of plugging or unplugging my Firewire connections while either the computer or the Fireface is powered, since there have been reports (right here in this forum) of damage caused to the Firewire chips located close to the connection points.