First, spear hafts: These were usually made by any of three methods:
Wood as grown, from coppiced poles or saplings. Coppicing is when trees are pruned near the ground to encourage the growth of long straight new shoots: https://commons.wikimedia.org/wiki/File:Coppice2.png which provides a renewable source of hafts. The coppiced pole or sapling can be debarked, and any small branches or lumps shaved off with a draw knife or plane, and the haft can be sanded if desired. Both coppiced poles and saplings are usually tapered, which is often taken advantage of.
Split timber. If the wood is too thick to use as-is, the wood can be split, and the split pieces rounded using a draw knife or plane.
Laminated hafts. Strips of wood or bamboo can be glued together to form a composite haft. A round core can be used, with strips glued around it. A laminated haft, if properly glued, is very resistant to splitting.
All of these methods provide a haft with the grain running continuously along it, which is important for strength. If wood is sawn and then rounded on a lathe, the grain often runs off the edge of the haft, and the haft can easily break along the angled grain. Coppiced poles and split wood aren't always straight - if close enough to straight, they can be used, even if the spear won't be quite straight. Otherwise, the wood can be straightened by heat-bending, using either dry heat or steam. An example of straightening spear hafts:
If the spear is intended for throwing, the balance should be forwards, for stability in flight. In this case, the thick end of a tapered haft is usually used as the forward end of the haft. For a spear that will be used for hand-to-hand fighting, especially if it is long, it is often better for the balance to be closer to the butt. In this case, the thick end of the haft is the butt.
Second, lathes. Lathes are over 3000 years old. The early evidence is mostly from art; there are also wooden items that appear to have been turned on a lathe. The earliest lathes themselves haven't survived, but the depictions in art are similar to pre-industrial wooden lathes used into modern times in many parts of the world. The lathes are typically driven by a cord wrapped around a pole or the workpiece. The ends of the cord can be pulled by an assistant, with one end held in each hand. The ends can be attached to a bow (similar to the operation of a bow-drill); a bow-lathe like this can be used by a single worker. A treadle can be used to pull the cord, allowing a single worker to use both hands for shaping the workpiece, and a foot for powering it.
An ancient Egyptian lathe (with a vertical axis):
Ancient Egyptian bow-lathe:
Treadle-powered Medieval European lathes:
Modern traditional Japanese lathe, hand-pulled:
Modern Indian bow-lathe (source):
Traditional lathes in action: