The mechanical speed change motor is 3 phase. So you are essentially bypassing and rewiring the whole lathe and mechanical speed adjustment when the goal was to run the lathe on a single 3 phase source. By not using the mechanical speed drive you loose the mechanical advantage, the Hp of the stock motor will fall off linear below the base speed and the lathe will be gutless at low speed. The carriage drive is run off of 2 legs of 240VAC and stepped down to 110VAC via a transformer, as least on the schematic I saw, the carriage drive is DC. So you may need to replace the transformer.
The stock motor is 1.5 Hp 3 phase, so you will probably need a 30A single phase 120V source, the Invertek 120V, 1.5Hp 3 phase 240V ODE-3-210058-1042 species an input current of 22A. They run around $320.
Seems like a poor patchwork of choices unless the lathe is dirt cheap, or you could use no motors and just add a foot pedal.