The machine can now be controlled by fingers!
I revamped the breadboard circuit (3rd time) and got all 4 axis's working using an Arduino Mega 2560 and four L293E motor control chips. 8 of the analogue ports had to be used as digital ports, which was fine. The chips themselves are really easy to figure out and the connections are essentially symmetrical in 2 dimensions.
The four push buttons, bottom right, allow any particular motor to be turned on whilst the others remain off. The potentiometers allow coarse and fine control.
Next job - extend the main upright frame upwards by about 50mm to allow the needle to be attached. Then connect the vacuum tube and test it out!