Getting back to the original question the issue boils down to a sensibly economic way of getting a suitable speed range. Assuming the three step Vee belt pulley already fitted to the lathe is retained and that a matching pulley can be found for the countershaft its fairly easy to evaluate what is needed when a VFD controlled motor is used.
Typically a three speed and back gear plain bearing lathe of that sort of size and similar pre WW2 to early post WW2 era would have a specified speed range covering 600 to 650 rpm. Speeds would probably fall somewhere in the 40 to 700 rpm bracket split into maybe 200 to 700 in direct drive and 40 to 150 in back gear. Actual speeds on offer varied with make but, so far as I'm aware, those ranges cover most machines.
Consider a pair of three step V belt pulleys as giving spindle speeds of double, equal to and half the countershaft speed. So for 700 rpm output the countershaft speed needs to be 350 rpm and for 40 rpm it needs to be 80 rpm. A 4 to 1 range which is technically possible using a VFD with a single speed drive but likely to be somewhat unsatisfactory in practice unless a relatively large motor is used.
It seems to be generally accepted that a VFD controlled motor speed range of ± 1/3 rd nameplate speed is almost invariably gives more than adequate performance for all normal purposes using a motor of appropriate power for the job at its nominal nameplate speed. The further you move outside this range the more likely it becomes that compromises in work capability or motor size will be needed. Home workshop types generally don't work their machines that hard so odds are the compromises won't bite too quickly but some engineering calculations to verify things may be needed. Usual issue is lack of power at lower speeds as VFD drives are essentially constant torque at lower speeds so actual power falls when running slower. Belt and gear reduction drives give higher torque at lower speeds. Normally dealt with by using a larger motor and offsetting the speed range so there is more variation at the high end that the low end. Larger motor means larger VFD so it all costs more.
In this case I'd use a two speed belt drive from motor to countershaft allowing a smaller motor to be used whilst remaining within the "always satisfactory" ± 1/3 rd nameplate speed range. Consider a VFD controlled 900 to 2000 rpm from an ordinary 1440 (nominal) rpm motor. Using figures from third paragraph top countershaft speed of 350 rpm at 2000 motor rpm needs 5.7 to 1 reduction and bottom countershaft speed of 80 rpm at 900 rpm needs just over 11 to 1 reduction. No point in getting hung up on exact numbers. Round numbers of 5 to 1 and 10 to 1 will do just fine. Just tweak the VFD control bit further. A 1/3 rd HP motor should do well enough.
Larger pulleys can be expensive to buy. Scavenge the drive shaft, housing and pulleys from an old spin drier to use as primary input drive giving a substantial speed reduction with a set of smaller pulleys up to the countershaft proper to give the two speed secondary. Use the spring loaded tensioner pulley off a car auxiliary drive belt system to tension the belt so no need to faff about when shifting speeds. Nice if you can use same belt for both speeds but two are no great hardship. Two belts worked fine for me once.
To an old school types like me (and Hopper) a VFD system sounds as if it ought to be expensive technology when compared to old school belt, pulleys and single speed motor. But if your scavenging sources are limited costs are probably similar these days.
Clive.