I have used Dado trunking
Trunking's not the problem right now. I'm sure it will be once I've dealt with the more pressing problems. At that point, I'm going to look for some discreet (thin) cable trunking that I can use for the run (near the ceiling) to the rears. If I paint it the same as the walls (which need repainting anyway), it'll probably blend in pretty well. Or I could just install some plain coving, I guess.
I hope that some of this may be of help to prevent or correct a problem.
Thanks for those tips, Ross. I'll try to bear them in mind.
For the moment, I'm going to move the DiskStation and the file server (Acer R3610) back to the hall cupboard. That'll hide quite a lot of the mess. Conveniently the VESA mount for the Acer can be fixed easily to the side of the cupboard, so it's well out of the way.
As far as the AV stuff goes, I'll look into building a thin false wall behind it (thanks for the confirmation, Bruno). There's not a lot of depth to the shelf, so my options are limited.
Of course, once the kids are a couple of years older, I can just put everything in a floor-standing cabinet. For now, though, it has to remain out of their reach.