Bite them, tear them or wear them down with manual tasks I imagine.

Flint could make a blade sharp enough to carefully carve your nails back if you really wanted

If you're careful you can bite a spot and then use a different nail to break the longer part without much hassle It's a lot less destructive to the teeth and has the bonus of smoothing the nail

Rough stones are easy enough to find if they needed to file them but if you've ever seen the hands of someone who does a variety of hard manual work every day it's more likely their nails were just worn and broken naturally with the odd bit chewed off.

> How did they cut their nails

They didn't. Probably like felids don't cut their claws.

No need to cut them if you occasionally idly rub a rough rock against them, filing them down before they can get long.