The antikythera appears to have been a form of Orrery or Planasphere not a clock.
The Greeks and the Romans had saws and files. They had dividers and a Mr Euclid to help them. They knew about screws (Mr Archimides) and rivets (old Doggynes).
I believe Heron of Alexandra was employed at the temple of Zeus in the Department of Applied Miracles. He built ‘slot machines’ which dispensed your ‘libation’ if you put a coin in them. He also seems to have built fully automated puppet shows.
To make the gears they would have done it my hand. There have been several descriptions of how to make a dividing plate in M.E. using lengths of tape with holes in them.