Hi Phil,
Good score on the boxes of bits
A single, variable drive's possible in a lot of different ways, but not all are ideal for a few reasons:
If you go for a DC motor and controller, there's an issue with running for long periods at low speed (or stalling the motor), as most use an internal fan for cooling - low speed, low airflow, less cooling! I see the consequences of this quite regularly, motor armatures with partial short-circuits (knackered) due to overheating the winding insulation, DC controller boards that have been hit with excess current (also knackered) by the partial short-circuit. This is the most common cause of failure with permanent-magnet DC motors
If you go with a DC motor and controller, I can recommend the KB Electronics boards - they've proved pretty reliable if not abused.
A 3-phase motor and VFD, there's the constant-torque characteristic, so reduced power at low speeds, and most motors get unhappy at less than 20Hz in and start to "cog" (the rotor "steps" between poles rather than moving smoothly) - there's also the cooling issue again.
Single-phase motors can't really be speed controlled, so they're out!
If you go with a variable-speed motor, it's worth adding a separate fan for cooling - my Holbrook has an irreplaceable motor (1950's 3-speed 4-6-8 pole 3-phase the size of a dustbin!) so I've grafted a big 240V fan on one end and have my VFD set up to switch it off whenever the motor frequency goes above 30Hz (and so whenever the motor's stopped, to cool it after use - until I power the machine off), this has worked so far!
If you're willing to go to the bother of adding a mechanical variable-speed drive, you don't have these worries to such an extent - this could be as simple as using the step pullies you already have and using a VFD or DC controller to vary the revs +/- 50% to avoid frequent belt-position changes, or you could look at moped transmission variable-ratio pulleys with an actuating mechanism of your own devising (Rivett 1020 lathes and CVAs used this method, the Rivett with a handwheel, CVA with motors to rotate a worm and wheel and adjust the pulley "spacing") - this lets you have continuously-variable speed from a single-speed motor with the advantage of increased torque at low speeds when you need it.
From the look of the pics, your machine has a back-gear for really low speeds, you'll need to use this!
Just my ha'pennorth,
Dave H. (the other one)