I had another play.
I was trying to get some stuff off the old laptop, which chooses whether or not to work on any particular day, and whose display backlight is on the way out.
I persuaded it to boot, and to connect to my network, but the desktop didn't see it (via Windows Explorer). It thought the "new" laptop was still connected. I left it to go for supper.
When I came back, the "new" laptop was gone from the Windows Explorer panel, and the old one was there. The network display worked correctly on both!
I think there's a "timeout" function for stuff connected to the network, and Windows doesn't pick up changes for several hours. I'll see if the display works tomorrow...