I am suprised nobody has mentioned the old method of polishing pivots on a block of hardwood with the arbor held in a pin vice! Obviously this is for clocks only as watch pivots are far too small. Jacot tools come up on e-bay quite frequently although the majority of them all seem to have something missing. There are various plans knocking about to make your own jacot attachments for lathes, the only problem with that is you need to have quite accurate machinery in the first place to be able to make a useable tool. It is quite frustrating when you are first starting out as you start to understand the importance of certain tools and then you see how much some of them cost 😯 but in most cases there are other methods of doing things. The problem with watch pivots being so small is as far as I am aware you either need a jacot tool or the lathe attachments. The only upside is you only need to buy the tool once, for if it is looked after it will remain with you for the rest of your life.
Good luck and let us know what you get