I'm talking way back. What was the solution to this? Was it even practiced?
As we're getting into pre-history, you should consider a X-Post to /r/AskAnthropology for this question. We have anthropologists who post here too, so they hopefully will see this, but you may have better results there.
I only have a very rough idea of places to look or avenues of inquiry you could begin.
Not saying that this is the answer, but when I was growing up, I had no access to nail clippers. I would use my canines to very carefully bite a consistent seam, sort of like how a wedge works on wood that you hit with the back of an ax. I wasn't taught, but I'm not so brilliant that I think this method is beyond anyone else.
If that were the ancient method, you'd need a preserved corpse from a non-tool-using society to verify. You'd need someone who knows what a carefully bitten nail looks like ( surprisingly few bite marks, FYI ), to look it over.
Relatedly, how do chimps and other primates clip their nails? If what you're really asking is "How do people without tools clip their nails?" then you might look at some of our relatives.
I know people in modern times who will only use scissors on their nails, and scissors ( I don't know this as fact, you never know with inventions ) SEEM very likely to predate nail clippers. So if you're asking what was the predecessor to our modern pinching clippers, you might want to look at and date a variety of ancient shears, and also go see if anything resembling clippers were found and when they appeared in historical record.