An update:
I installed a 2nd copy of win2000 on another partition, and installed an older version of Nero (6.3.something) under that. It worked perfectly.
Then I removed the latest Nero (the one with the fancy picture-based control panel, for idiot users) from the main installation and installed the older 6.3 one, and it works!
A mystery cleared up.
Thinking about it, it was the latest Nero software which the troubles started with, but it wasn't noticed because most of the time I was writing CDs, and they were always fine anyway.
It seems that one can use Nero with a dual-layer DVD writer (Plextor in this case) even though the program itself predates dual-layer DVD writers. It correctly reports "writing side 1, writing side 2" etc - I guess these messages may be coming from the drive itself...