Hi AA, sorry about the late reply - been sunning myself by a pool for a couple of weeks!
I agree it could be a dynamic balance problem, although static balancing may be enough to cure it - I know motorcycle wheels are only statically balanced, and the pulleys have a similar "form factor"? To do a good job of static balancing you need really low friction bearings to rotate the parts on - knife edges or similar will do as long as they're dead level and sharp and the temporary shaft's true and a good fit in the pulley bore.
A *properly balanced* lathe should be smooth at all speeds - OK, mine has 2 tons of cast iron to absorb and vibration, but I can just about balance a coin on the headstock at anything between 5 and 2200RPM as long as I don't have any of that awful work stuff in the chuck

The big old BTH motor was specified as dynamically-balanced though, back in 1956...
As a thought (possibly showing my geek roots here...) re: discovering the source of the vibration, there are some free spectrum-analysis utilities on t'interwebby thing, using a cheap microphone as a pickup would let you look at what vibration frequencies you're suffering from - if this matched (e.g.) the spindle speed or the speed of one of the pulleys / the motor, this could point you in the right direction?
Another thought, the "KB-alike" motor speed controls don't produce actual smoothed DC, instead they provide chopped "raw" DC, consisting of a series of pulses at 100 pulses/second (50Hz AC input, 120/second on 60Hz) - if this frequency "beats" against the commutator period or the motor rotational speed it could cause torque variations and hence vibrations at specific motor (and so spindle) speeds?
One possible fix if it's a speed control / motor issue would be to add a SUITABLE capacitor across the motor inputs or a line choke in series, bearing in mind that there will be as much as 220V DC on it *and it reverses polarity with the motor direction* - the cap' would need to selected for a minimum of 300V working voltage, *non polarised* and cope with perhaps a 6A ripple current - all I can think of is the cap's used in high-power (think heavy-metal-band PA system!) loudspeaker crossovers....
A choke could be made from a scrapped power transformer with a secondary winding rated for the current required (say 4A for a 450W motor, up to 15A or more for one and a half Chinese horses on 220V, double the current rating for 110V) with the primary winding *thoroughly* insulated?
The pots on the speed control board are normally (working from the left-hand edge): Max speed; Acceleration (further from the edge); Min speed; a gap; IR Comp; CL (current limit) - speeds and acceleration are pretty obvious (although increased acceleration can make a crackly speed control's effects worse), CL *should be left alone* unless you have a suitable ammeter to set it up, IR Comp controls the current boost with increased load - set too low and the spindle speed varies too much with load, set too high and the speed will "hunt" up and down or surge. *All* the pots are way too sensitive on these boards, I think because they've tried to get a wide range with a single-turn pot!
Just my ha'pennorth,
Dave H. (the other one)