Welcome!
Not into such small sizes as used in watches, but am sure that you will find folk on here who are knowledgeable and skilled in such matters, and can help you.
Some one is bound to say that if you are making your own lathe, it will need to be made to a far greater accuracy than that to which you want to work..
Having said that, watchmakers lathes seem to be pretty simple compared to some of the larger machines that we use.
Maybe the difference is in skill and the wish to work to a very fine dimension, rather than to a size that allows the device to work..
Howard