To compose my previous email I checked various things, but to no avail.
Subsequently, I've tried removing the homegroups and setting them up again.
I now find that I'm further back than I was.
I did the following:
I'd already tried cycling homegroup through Home / Work etc on both machines.
I decided to exit the homegroup on each machine - which meant that there wasn't one.
I then set up a new homegroup on each machine as before, and checked all of the settings, such as self-discovery etc.
So I should be back to where I was a couple of hours ago.
Unfortunately I now have the following differences:
Whereas the laptop previously appeared on the full network map as seen by the desktop, it no longer appears.
The laptop still can't see the desktop.
Whereas each machine could previously, using Media Player, play music which was resident on the other, this is no longer the case.
(Both machines access the web via the router, but neither can see the other.)