| I can't tell from the pic, but it looks like the nut is sitting in a slot, like a Fender? If it is, the thickness is going to a critical dimension.
In general, the critical dimensions for making a new nut are the thickness, the width and the height. That's assuming you'll cut the slots for yourself, and finish up the radius across the top.
But if you don't want to do that part yourself, the best thing to do is to take the bass to whoever does the job, and let them fit the new nut in place, and cut the slots. Personally, I wouldn't try to make a nut for someone else without having the bass in hand.
Ed |