Before you attempt to remove the nut, stack feeler gauges on the fretboard in front of the nut until the gauges are level with the bottom(s) of the nut slots. That'll give you a baseline measurement to aim for if you unintentionally destroy the old nut and can't copy it to make the new nut. Once you have this measurement, you can put the new unslotted blank in the nut slot, stack up the same height of feeler gauges, and mark a line across the face of the nut.
When I am removing a nut, I use a hobby knife blade to score along the front and back edges, and across each end. This helps prevent finish chips if the finish has glued itself to the old nut. Then I clamp the instrument on it's side (i.e. in playing position) and use a hardwood dowel and a mallet to tap on the end of the nut, pushing it down and out, more or less "sideways" relative to the neck. This helps prevent taking chips of wood out of the front/back face of the nut slot. Once the nut is out, inspect the slot to see if there is any residue and clean it out with a file as needed. Then you can install the new nut. I use one very tiny dot of yellow wood glue to hold the new nut in place. You really just need to tack it there, using too much glue or a glue that's too strong is a big mistake as it'll make future removal impossible.
Speaking of too much glue - if your nut has been glued in place and you can't tap it out sideways, usually the best course of action is to clamp the neck in a miter box and cut a slot right down the center of the nut. Cut until you're just about to hit the bottom of the slot. Then you can collapse in the front and back faces of the old nut and remove the pieces without worrying about damaging the slot. It's very easy to do major damage otherwise.
If your old nut is perfect other than the one low slot, you can use it to make the new nut. Clamp the new and old face to face and use a sharp pencil to transfer the slots and the overall profile to the new nut. Then you can cut the new one to shape. I rough in the slots with a saw and then finish them with files. If you don't own fancy nut slotting files, you can use sandpaper wrapped around an appropriately sized drill bit. Roughing in the slots with a saw will speed up the process vs trying to file the entire slot from scratch. The front edge of the slot needs to be crisp to give the string a good point to break over, and the slot should slant towards the back of the nut to keep the witness point right on the front edge.
