Andy,
I turned the lace bobbins by hand. That was long before I got involved in CNC stuff. In fact in 2004 we spent our last ever weekend living in the UK attending a course on bobbin turning and one of the first household appliances we bought in the new country was a lathe! In our early years in Australia C used to have an online shop selling bobbins nd other lace-making supplies, some that I made here but most were imported from the UK made by Stuart Johnson (who ran the course we attended along with David Springett) or Chris Parsons.

Hand turning wood down to a diameter of 3mm takes a steady hand but once you get the knack it's easy enough with the right timber. Mass production requires a well thought out process and the right tools but nothing expensive or complicated. If you really want to look into making a 4th axis and producing bobbins I'd be happy to give you the benefit of my hand-turning experience. A 4th axis for such light work would be pretty simple. The main design requirement would be a fast turn-round time for swapping out pieces.