I fixed up the dead pickup and reassembled it with hot glue (thanks @ICBM).
I've replaced the under-pickup foam, as the original had perished. However, it seems that the neck pickup was screwed right down and actually under the pickguard as the cutout is too small and the screw lugs are in the wrong place. See pics.
Pickup on top of pickguard:
Pickup underneath pickguard:
I'm thinking I might get a new guard as the previous owner bodged the fitting of the thumbrest - it was screwed on in the wrong place (see additional holes at the bottom of this picture):
Any idea if I can get a pickguard with correct-sized holes?