Well, after some serious fettling, it's looking a bit more promising...
The tyres, while hardly worn, are thoroughly past their sell-by date and disintegrating on the inside. Front tube has several patches and a slow leak. Replacements ordered.
Trued both wheels.
Attempted to get the freewheel off with a view to giving the bearings on the rear hub some TLC (it's making some seriously horrid noises), but, as usual with freewheels, failed. Re-greased the side I could get at and adjusted, which improved things somewhat.
Dug out some decent V-brake blocks, determined my spare brake cables wouldn't fit, ordered a longer one.
Dug out an adjustable stem, which brought the bars back and up, but not quite back enough. Hmm.
Removed a link from the timing chain. That made it too short, so I put it back again. Added some washers to the tensioner mounting to make it mostly okay.
Tweaked the gear adjustments, determined the gears are scarily roadie (42-28 is as low as it gets).
Tweaked the disc brake to stop it rubbing.
Adjusted both saddles to good effect. Sheared a seat-post bolt.
Raised the stoker bars.
Stuck some SPD-compatible pedals on the front.
Had a few short test-rides, one with stoker on board. Nothing broke. Nobody died. My riding position is now 'uncomfortable' rather than 'insane'. Stoker still has a sense of humour.