Here's the machine ready for its first test:
The XY stage seems to move smoothly with very little play. The gearing produced by the threaded rod seems plenty. It does hesitate sometimes - I'm not sure if this is because the stepper motor output is set to pretty much minimum causing a stall, or if something is catching or slipping.
The Z stage has two problems. Most severely, the frame is being flexed excessively. This is partly due to too a corkscrewing action in the threaded rod (unsure of cause) and partly due to a lack of rigidity in the frame. Less severe is some play in the sliding frame, which might be fatal to useful printing.
At this stage it seems sensible to redesign the Z axis.