A good classical guitar should NOT have the nut glued to the wood. Gluing the nut is common in cheap guitars. Usually, I recommend NOT to touch the nut. In most cases (it may not be the case of a cheap guitar) the action can be handled by modifying the SADDLE. Try the saddle first. Sand it down on top of a leveled hard surface (such a the kitchen granite, or a thick glass) where you have placed a piece of sand paper. To check the height of the nut, press the string in the middle of fret 3 (that is between fret 2 and 3) and notice the distance between the string and the top of fret 1. This distance should be about 3 tenths of on millimeter (almost nothing). If it is more than that, the nut could be adjusted without need to remove it. Get a file similar to the metal files used for filing the nails, but with filing edge (nut files), or if you do not mind and can do it carefully, you can get a credit card and wrap a piece of sandpaper (100 or 150 grit) to act as a nut file. Place yourself behind the head, separate the string out of the notch and file a little bit. Check again by pressing the 3th fret. FIle again if needed, until satisfied. Do this for every string. If you have a good eye or good measurement devices, you will notice that the bottom of the notches in a nut are not aligned or forming a line parallel to the fretboard surface. Instead, the go from lowest to highest from high E to G, and then from low to high again from D to low E. I hope this works.
Pepe, el Nino de Las Guitarras
"It is not enough to just ride this earth. You have to aim higher, try to take off, even fly. It is our duty".
â€” Jose Yacopi, Argentine Luthier