OK, an update.
new tyres, new track rod ends (for the MOT), no bushes failed the MOT.
I looked into the preload - driver's side was a bit slack so removed a shim from it. Passenger's side seemed more or less right, might be a tad slack but not so much so that the swivel flops over to one extreme on it's own. I left this one alone.
It still does it. Since fitting the tyres, it doesn't go into a terminal wobble, but it's still very definitely there.
I happen to have some new dampers around, may try changing them next. However, anyone got any other suggestions?
Steering damper may not be up to spec but it's not shagged either. I don't have a new one of those to hand to compare, although I do have what seems quite a good used one about. Might be worth swapping it to see if it makes any odds.