OK, FWIW, here goes.
The laptop was upgraded from Win8 to WinX. The Logitech Mouse that is being used with it, the M510, is not supported by the Logitech Options software that Logi now uses for their (pointing) devices. I did not know this until I went to install Options - after installing (thanks for wasting my time, Logitech!) it pops up a message saying that the connected mouse is not supported, and that I needed to install SetPoint.
The machine never had SetPoint installed - and never needed it. So , I uninstalled Options and continued to troubleshot other issues on the laptop.
When the machine was finally ready for me to make another attempt, I let AU start its thing again. Sure enough, on first boot, it hangs. So, I unplugged the receiver (yes, it is a unifying receiver, probably a very old one), and forced the laptop off an rebooted, it rebooted back and continued to install the AU.
FWIW, my own laptop *and* my desktop both have Logitech Unifying receivers. The laptop has one from the Logitech Performance Mouse MX, and the desktop has one from the MX Master. Neither machine has ever had a problem when rebooting during the WinX upgrade installation process. So, I didn't suspect that the one in my Dad's computer could be a factor at all.
Looking at the hardware, I also see that the unifying receiver from my Performance Mouse MX is older than the one that my dad bought, in terms of actual purchase, but that his hardware version is older than the on with my Performance Mouse MX.
IOW, he probably bought it from a discount retailer to 'save money'.
I've plugged in a generic onn wireless mouse and it is working perfectly fine with the machine.
Thanks again for the tip, rpo. I gave you rep on that one - I'd never have thought to look at the unifying receiver.