Do nuts already come with the slots cut in them...
Some do, some don't. If they do have pre-cut slots, you'll want to be sure to get one with the right slot spacing for your guitar.
...if so, is the radius set by the manufacturer, or is that step left to whomever is doing the replacement?
If the slots are pre-cut, there is a already a radius to the slots. But you can file them to whatever radius you like.
Ignoring the slots for the moment, the top of the nut will likely also have a radius cut into it. You'll want to have that match the radius of your fingerboard.
Some nuts have a radius built into the bottom. These are for guitars that have a radiused nut slot. if your guitar is like that, you'll want to match that radius exactly.
All of these things are adjustable with the right files, some sandpaper, and some practice. And they interact with each other. For example, slotting files will lower one string at a time; sanding the base will lower all six strings at once. Sanding the base
at the wrong angle can cause a whole range of problems. And there's an art to filing the slots just right, so there's a clean takeoff point, and the far end of the string dresses cleanly and freely toward the tuning peg, and the string doesn't sink down too far into the nut, and...
That's why having a pro do the work will cost you more than a few bucks. Each nut is a little sculpture, carved precisely for your specific guitar.
Bottom line: You're not likely to find a pre-made nut that you don't have to do some work on.
