Hello again Chris,
I can add some information now on how the gearbox works. The final gears on the right hand side of the gearbox are 32 T on the cone shaft, which is permanently meshed with a 16 T gear on the layshaft above it. That gearing provides the ratio change needed for the coarser pitches where the leadscrew is made to turn faster.
The actual ratio changes of 1 divided by 2; 1 to 1; and 1 multiplied by 2, are provided by the top selector handle and brought about by dog clutches in the gearbox. The eight cone ratios are made via the sliding selector on the face of the gearbox. So the gears you are concerned about are NOT accessible by the motion of the gearbox controls.
The more I think about your problem, the more I feel sure the woodruff key linking the drive from the outer 72 T gear to the gearbox input shaft is missing, clamping grip from the nut holding it in place is not enough to allow for carriage movement.
One final forensic question. Does the rest of the gear train from the headstock down continue to turn happily when you engage the carriage half nuts? If so, I think that confirms my diagnosis.
Regards Brian