As Ken stated, the speed controllers for rotary tools (dremel, Foredom, etc.) and routers are ONLY for use with universal motors. How do you know if a motor is universal or not? If it has brushes and screams like a banshee, it is a universal motor. (Note: brushes are usually the best tipoff, but there are some very old repulsion start induction motors that also have brushes. Fortunately, you're not likely to find such on a modest lathe as the typical repulsion motor would be more massive than the lathe.)
Typically, with lathes, the motor will be either DC or Induction, with the vast majority being induction, which are either 'speed controlled' with differing pulley ratios or variable frequency drives. Pretty much as a rule, those lathes equipped with DC motors (which would otherwise work with your speed controller) already ship with a variable speed controller (and you don't want a speed controller feeding another speed control circuit).
On the other hand, if you do come across a lathe with a universal motor (a cheap HF?) with an appropriate amperage rating, then you could safely (at least with respect to the controller) use your speed controller so long as the lathe also lacked a speed control or soft-start circuit. This said, as someone who has never used a lathe, I'm not sure how safe a lathe with constantly varying speeds might be since I would imagine the speed variations would also alter the aggressiveness of the cut in possibly unexpected [and unsafe?] ways.
Even between universal motors, your speed controller will have a maximum design amperage (and a practical minimum below which it regulates speed poorly) which can not be exceeded lest you destroy the thyristor.
