Based on the old wooden wagons I've seen, lathes played no part in their construction. The spokes on the wheels had been spokeshaved (gee, maybe that's the etymology of that tool's name?) rather than turned on a lathe. The wheels as a whole were pieced together by hand so they'd be round automatically (i.e. spokes and rim segments were all the same length, held together by an iron hoop that just took shape around them when cooled).

Sure this is just one example, but it is what comes to mind given TFA's subject matter. Without access to iron-working, I suppose a less durable rim material like e.g. hemp could be used, which would introduce some more complications involved in tightening. If solid rather than spoked wheels were acceptable, an out-of-round wheel on an axle can be marked with a piece of chalk-rock held in place while spinning the wheel, and then abrasive stones can be used to true the wheel to the chalk mark.