Gee wiz, that did it!
I shut down the Windows Firewall and reinitialized the NSA and at the end of the firmware installation, it was giving a message about rebooting, but while before it was not doing anything, now it actually did reboot. After it was done rebooting, I could access the web UI and all!
I remember reading in the Troubleshooting section that UDP port 50127 had to be allowed through the firewall for the NSU to function correctly. Still, there was no mention of storage needing to be attached, so that got me stuck for a couple of hours... Couldn't imagine not having enough flash to hold the firmware!
Anyway, now it's back online, so thank you for your prompt response!