For what it is worth I will make another suggestion....
Were it me in your situation, I would not worry about nut files and just take the extra height off the bottom of the nut by useing a flat surface and some sandpaper...(The more truly flat the surface the better. A piece of glass (or glass cutting board), some tiles, a level, cast iron table, etc.....get creative I am sure you have something that is flat)
Go very slow and tell yourself before you begin that you will likely have to slacken, make changes and bring it back up to pitch many times before you get there. (no matter which way you choose to do it)... The best way to screw up is to think it will happen in one or two tries or be in a hurry.
-Get some precise measurements (with the instrument(s) up to pitch) so that you know approximately how much you need to take off. DO NOT try to take it all off in one go!
-Slacken strings enough to remove the nut (count how many rotations you turn the tuners as it can help you get close to pitch quicker)
-(Optional) make a pencil mark on the nut for the amount you want to remove. then take the pencil in your hand, put your middle finger on the bottom face of the nut while laying the sharpened point on the mark. Use your middle finger to keep the line the same distance from the bottom face and slide the nut across the pencil to strike a line that is level. Do this on both sides of the nut. Some people have a hard time using the finger to guide a line but it is really easy for most with some practice so feel free to get some wood with a flat edge (watch for splintery wood) and test it out. If you cannot get this to work it's ok, just go slow and do not take off too much at once. If you do get it to work do not take it all off at once anyway as it is just a ballpark and if you do not want to take too much off.
***I hope this makes sense. I am just trying to describe an easy way to strike a line by using the edge as your guide so that you can visually see your line is still straight while sanding. ***
-Run the bottom face of the nut across the sand paper (120 - 220 grit will work nicely) just a few strokes at a time, then rotate the nut 180 degrees and do the same number of strokes (this will help with the fact that there is always uneven pressure and help keep things level), just try to stay consistent on pressure and technique. I find that the more pressure you use while sanding the more out of "true" you get. So go easy on it and take your time. Since it's you first time doing this only go about half of what you think you need to take off. and check you progress each time....once you sand it off it is not going to go back on.
-Reinstall the nut & bring up to pitch (remember how many rotations you slackened? It will help you get close before turning on the tuner)....
-Check both feel and measurements...
-Repeat until you get where you want to be.
The key here is to not take off too much and not get too uneven or out of square.
Also know that the numbers from your measurements are a guide. If it feels fine and is .5 mm more that is fine no need to push it and risk going too low. Feel is always more important than numbers when it comes to action.
I hope this helps....
You got this!