I only have one lathe, and it is a big heavy thing with a 1 9/16" headstock hole and a 48" bed. To make fine small screws, I just chuck up a zero Jacob's chuck in the headstock and Bob's your uncle. I don't usually make all the screws for a lock, but have made replacement screws when a frizzen screw has become worn or damaged, a sear screw that is sloppy or doesn't bottom on the plate before it squashes the bridle and sear, etc. On occasion, I will make a full set for a lock set from The Rifle Shoppe, or such.
I've made some nice round headed lock 'nails' out of 1/4" carriage bolts, by turning off the square shaft and reducing to 3/16" then rethreading 3/16 x 24 or #10 x 32. They have a perfect large head and you cut your own slot.
I use mild steel rod, usually, and case harden the screws. I always case harden lock 'nails' and tang screws. Otherwise they quickly look like originals from ham fisted fools.