If considering using a variable speed DC motor on the lathe, first thing that you need to determine is whether or not the drum switch that you want to use is suitable for 3-Phase and DC motor operation. To determine this, fasten the switch to something either bolted to a bench or somehow fastened down so that you can operate the switch one-handed (not having to hold the switch down with one hand while operating it with the other hand). Operate the switch to one of the two ON positions (call it Forward for the moment). Now try to change the switch to Reverse in one continuous motion (without momentarily stopping in OFF). If you can do this, do NOT use the switch on anything except a single phase motor. It is NOT safe for use on either a DC or a 3-phase motor.
The reason that this is so is that a single phase motor when running in either direction (assuming that the centrifugal start switch is working) will have the START winding disconnected by the centrifugal START switch and throwing the control switch directly from FORWARD to REVERSE will do nothing except momentarily remove power from the RUN winding(s). The motor will continue to run in whichever direction it was running before you changed the switch. The DC or 3-phase AC motor, on the other hand, will create much hate and discontent as it tries to go from full speed in one direction to full speed in the other without being allowed to momentarily stop before starting in the other direction. A rotary, lever or toggle switch that is made to operate a reversible motor has a mechanism built into it that basically forces you to remove your hand from the bat/lever/knob while it is in the OFF position before you can throw the switch on over to the opposite setting. Determine this before you even bother to try to figure out how to wire your drum switch. If the switch came from Atlas, it is probably OK.
We have switch wiring diagrams for a lot of motor switches. but it isn't worth looking it up until you have tested the switch mechanically.