Well, that's rather odd.
Put the latest BIOS revision on a floppy, put it into the new PC, and loaded the EzFlash2 program via the BIOS. It showed two drives, A and B. It found nothing on the floppy even though I could hear it being accessed for a short while. I then tried to access 'B', and surprisingly I heard the NEC rev up, and the directories of the Windows CD were shown. It seems there's no communication problem then.
So, having not been able to update the BIOS to the newest revision, I'm unsure what to do now. I noticed that the clock seems to have gone to 00:00, and has now gone up gradually as I've been using it today.
Any ideas?