Hi Bob,
Make sure that you buy a watchmakers and not a small modelmakers lathe. There are a few good makes to choose from, Boley, Lorch, Levin, Pultra and Rolls Royce of lathes Schaublin. Whatever you go for make sure that it is in excellent condition and with as many collets and attachments as possible. The reason I say this is to purchase the attachment separately will cost a lot of money.
Lathes such as Unimat are excellent for modelmakers, but do not have such a comprehensive selection of collets and attachments that you will require for doing watch repairs.
I have a Boley Lienen WW bed lathe and attachments. If you check out this http://lathes.co.uk/leinentradwatchmaker/ and http://www.lathes.co.uk/leinenmodern/ you will get some idea of the sort of things available. If you go to the homepage of the link, you will be able to look up information on just about any lathe, both large and small.