Without knowing the lathe in question, I would back up Chas's figure and warnings of difficulty.
It is not only the question of a 3 phase motor and an electronic box, but understanding the box, programming and wiring it up to some sort of control box.
A friend of mine got a Graduate for £50 from a school displenishment with an old 3 ph motor in it. I just happened to have a 1 to 3 ph convertor and thought that all that was required was to wire it in - please note at this point the male attribute of assuming one doesn't need to read the instruction manual ! Also note that I am an electronics engineer .... which also means that I don't need to read instruction manuals on this sort of thing.
Well there had to be a lot of Instruction Manual reading and consulting of Mr Google, the motor had to be dismantled and rewired ( Delta to star for the infomed), a control box had to be built, the wiring of the machine had to be gutted and reconfigured.
Are you beginning to get the message ? It does seem easy but (a) there's a lot of work and (b) it's not easy.
I'm quite happy to do it for anyone having been up the learning curve, but despite being retired I will expect to get paid for my time.
OW