Incredibly, you were right!
My computer was connected directly to my router via Ethernet, but there were some HomePlug type devices on my network. It appears my neighbour has recently installed similar devices and our home networks were leaking into each other over the power lines!
The DHCP servers on our routers were competing with each other so all of our devices kept switching from one to the other, causing the dropouts I was was seeing.
I unplugged my HomePlugs and everything went back to normal.
I'm quite surprised it's so easy for this to happen, partly because we live in two separate houses rather than a block of flats or something so I would imagine our power circuits would be more isolated, and partly because I thought the pairing process between HomePlugs was meant to prevent this kind of thing from happening.
Because I knew the PlusNet router was based on a BT HomeHub, when I saw the references to BT my assumption was that something had gone wrong with a firmware update and some old BT UI had become active. It never occured to me that I was actually looking at a different router!
I'll be having an interesting conversation with my neighbour and leaving my HomePlugs turned off, indefinitely if I can't find a way to better isolate them!