The use of specialized tools for nail care goes back to ancient Greece at least, where nail care was done by the same fellow who cut your hair. The tradition was imported to Rome.
To give you an idea of what kind of instruments were used, this object from the St. Albans Museum is listed as a replica of an ancient Roman nail-care kit, and it includes something that looks very close in form and function to a modern nail clipper. (EDIT: But which actually is not, see my below discussion with kermityfrog.)
I can find scattered references to Egyptian nail care tools, but nothing as solid as that. Getting earlier than Egypt and Greece, we'd be looking at Mesopotamia. I can find no specific references to nail-care tools of that region's civilizations, though some of their myths have characters removing dirt from beneath their fingernails with no mention of a specific tool. And it looks like it was customary to impress an unfired clay tablet with your fingernail as a kind of signature , so for at least some of the ancient Mesopotamian civilizations, for at least some time, it seems to have been customary to leave your nails long.
Oh, and I found a lot of modern sources offhandedly mentioning that manual laborers' nails wear down and break on their own, and 'manual laborer' describes an increasingly overwhelming majority of the population as you get further and further back in history. I can't give you anything so academically rigorous as a primary ancient-world source that directly says that, but have a photograph of the hands of a modern Bangladeshi rice farmer's wife.
So it looks like quasi-modern nail clippers are about as old as documented traditions of nail-cutting. Unless there's a cave painting somewhere out there that tells the story of The First Manicure. (EDIT: Again, I seem to be mistaken here. The Romans used a specialized sort of small knife: see this discussion for further info.)