I've never made nutmilk, but I'd guess the fine-ness of the nut puree could cause problems. You can put a coffee filter over the metal strainer (so that you push it down into the liquid) to keep particles out, but then you'll still have liquid trapped with the puree under the metal strainer. It'll also be kind of a pain to press down with the paper in there.
If you're okay with losing some of the nutmilk then it may well be easier. But then if you were okay with it, I'd think you wouldn't be so annoyed by the traditional method.
I guess if you use a LOT of puree relative to the amount of water, you could press it directly with the strainer, but even so there's simply a limit to how much you can press in a French press. If you're like me, you'd end up eyeing the puree at the bottom suspiciously, decide to throw it out, then find yourself scraping it into the nutmilk bag anyway and cursing the father of all nuts for making you do this.
Again, I really don't know on this one, but I'm guessing it's 6 of one, half a dozen of the other. If you're willing to let the French press tell you "that's enough!" then that method would be easier. If you're thinking you're going to get every last drop, you won't escape the nutmilk bag, and you basically need to decide if you want to wash your French press too.
ALSO, if you use the French press, you'll want to disassemble it and wash it thoroughly in the first place in case it's new for obvious reasons, and in case there is any coffee oil residue if it's been used for coffee. That coffee aroma is hard to escape.