The lathe chucks aren't compatible with the new ER spindles used on the mills. The threads are different, and an adapter would have to be custom-made; it's not a standard part. If you really want to do this, you'd be better off getting one of the old-style headstocks, with the 3/4" - 16 thread, which are still used for the lathes https://taigtools.com/product/100-00...3-4-16-thread/. But these don't take ER collets, which are superior to the old-style proprietary collets that fit the lathe headstock.
Also, a mill is not the same as a lathe. The geometry of the operations is different; so even if you got a tool post mounted to the bed of the mill, you'd have a very limited reachable area of the workpiece that could be turned in a vertical orientation before it would be pushed into the bed. Taig makes lathes as well as mills, and if you're already getting a chuck (which Taig makes) and a headstock, you're most of the way towards one: https://taigtools.com/product/micro-lathe-ii-k1019-kit/ Turning your parts on that would work much better than trying to get the mill to function as a lathe.