I use a very cheap needle file set. A Nicholson set of 5 is around $10. I just use the round one mainly. At it's widest end (even where the handle would fit into it if it had one) it's about perfect for E. Very close to the tip works for G. Get a file card to keep it clean and cutting.
According to Erlewine, the string slots should be around the radius of the string deep so the string can vibrate properly. Typically that's the case on vintage instruments. Today it seems they come from 'the factory' deep, elliptical and way too high above the first fret. The result is bad intonation on the first frets. I love to see strings standing proud of the nut and bridge saddles. They stay there even with aggressive playing. I use a smooth mill file to file down the nut after you get the slots deep enough and then the flat needle file to smooth it out, then polish w400/600 paper