The major thing that took time to work past was that the insulation that permeates the upper floor is covered in aluminium, and the wireless router that covers this end of the house is sitting on the other side of it from the Synology.
So we made a trip to Maplin, bought some Powerline* kit, a switch and a bunch of network cables, and moved his TV, Sky box, and the NAS over to hardlinks rather than wireless connections. Now it all just works.
The funniest bit was the insert in the switch - switches are ancient technology (particularly this one, which is 10/100MBit, because the Gigabit ethernet switch was larger for the same number of ports), and clearly this one had been sitting on a shelf for a long time. It came with an insert that was advertising BTOpenWorld business internet connections. A shiny new form of internet connection that could, apparently, give speeds of up to _2,000_Kbps, so long as the computer it was plugged into had 32MB of RAM, a 4-speed CD-ROM and either Windows 2000 or an Apple Mac running on PowerPC 601 or equivalent running Mac OS 8.6 or higher...
*Sends networking signals over the electrical cables. Incredibly handy if you can't run networking cables absolutely everywhere.
Original post on Dreamwidth - there are comments there.