How do I go about filling a hole that's all the way through wood, for a new nut?
The best way to deal with this is to drill out the holes to a larger size to match a piece of dowel rod. The dowel rods come in standard diameters so select one that is a bit larger than the hole that needs plugging. 

Cut off the dowel rod to a length that matches the thickness of the door or drawer front. Then use wood glue to completely coat the outside of the dowel and inside of the hole and insert the dowel into the drilled out hole. 

Use care to match up the outside surface of the wood with the end of the dowel so the surface is nice and level. Let the glue fully dry. 

You can now drill your new hole through the wood where needed for the new handle of pull. If the original wood is a soft wood like pine and the dowel was a harder wood like birch then it is advisable to use a drill guide to hold the drill bit position if the new hole needs to span the transition between the dowel and the existing wood of the door/drawer front.

Before installing the new handle or pull you can use some wood stain or paint to touch up the ends of the dowel so it does not show out too much. If you got glue all over the ends of the dowel and it dried there it will first be necessary to sand this away before trying to apply stain.