Take a look at this. That's a trapezoidal thread (stronger than a conventional V thread as used in normal nuts and bolts). It's 12mm outside diameter (a bit bigger than your 3/8" but it's the nearest example I found with only a quick look). The important bit, though, is that it has effectively 3 threads running round it. Each thread has a pitch of 6mm (one turn of the rod would make a nut move along 6mm - that's what pitch (well, theoretically you should say "lead" but let's not get too pedantic) means) but is only as deep as a 2mm thread. So this rod has 3 threads, so if you took a rule to it, it would look as if the threads were only 2mm apart, but that's because there are 3 threads each with the 6mm pitch. It's a 3-start 6mm pitch leadscrew. The really important bits of the specification are the diameter - that affects how fast you can spin the leadscrew as they start to whip as they get longer and if they are too thin - and the pitch, which is how far a nut moves for one turn of the screw. You need that for various things when you are setting up the control software.

I don't know exactly what that original leadscrew specification means, but there's nothing magic about the exact values. Something like a 12mm x 6mm pitch 3-start thread would do a decent job for the kind of machine you are talking about. If you could find it, then a 10mm x 4mm pitch 2-start would probably be OK and probably nearer to the original. The Zapp website has a bunch of alternatives that you could look at.