Take your pick.....
https://www.cousinsuk.com/category/pivot-steel
https://www.hswalsh.com/product/blue-pivot-steel-wire
http://www.m-p.co.uk/muk/parts/chap10/pivot-steel-0.61mm.-10pcs-no.72-0618007215.htm
https://www.esslinger.com/staff-and-pivot-watch-and-clock-wire-rod-assortment-blue-steel-55-1-93mm-12-pieces/
http://timesavers.com/i-24049935-30-pack-3-00mm-3-95mm-blue-wire-assortment.html
https://www.amazon.com/Assorted-Tempered-Pivots-repair-pinions/dp/B01N1SOLDF
http://redroosteruk.com/11-assorted-tempered-blue-steel-wire-pivots-clock-repair-pivot-pinions-staffs/
https://perrinwatchparts.com/collections/metals/products/pivot_wire_43_2115?variant=37875445135
Google "blue pivot steel" and you will find more.
This has been the basic stock for watch and clock pivots and staffs forever. It is a carbon steel that has been hardened and then tempered back to a dark blue so that it can be cut on a lathe. It is still quite hard, try and cut it with snips and it will resist, eventually giving suddenly and breaking rather than cutting. In the lathe though it cuts easily with either a tungsten carbide or HSS graver, although a HSS graver will need frequent touching up to keep it cutting with a good clean finish. Keep a hard Arkansas slip by the lathe for this. Let it get too dull and it will chatter, resulting in a poorer finish, and it will also cause you to apply more pressure to get it to cut. If you're not careful you will break off the work piece by pushing too hard. Don't try to cut it too quickly either, and don't expect long curls of swarf. Take your time and make dust. You will get a better finish and the HSS graver will keep its edge longer, whilst the tungsten carbide will be less likely to chip.It also burnishes well so don't try to cut the pivots to size in the lathe, leave them over size and finish in the Jacot tool.
This doesn't need any further heat treating when you're done, it already has the right balance of hardness and strength for pivots and staffs, especially if you burnish the pivots in the Jacot as this locally work hardens the surface giving the correct hardness to resist wear whilst the rest of the staff retains enough strength to cope with a degree of mechanical shock without shattering.
Practise and enjoy.