Well, I see at least 3 reasonable approaches here, and I'll add another. In one case I stained and finished all the pieces before assembly. For the nail holes I made filler using sawdust and glue, adding enough stain so it matched when dry. Test this first, as the dust will take the stain differently. For each hole, stuck some in and immediately wiped with a damp sponge. The finish prevented either the stain or the glue from sticking at all, and it easily wiped clean. Did several hundred nail holes this way.
After it dries, check with a raking light. You may find a few holes that aren't completely filled, and do a second pass.
One variation where you have a gloss finish is to apply only enough finish to seal, then the final finish coats after the holes are filled, so the filled holes have the same gloss.
Note: If you should end up staining after nailing, often a dark stain will cause the hole to be outlined because of the grain difference. Even nailing through tape may not prevent this, so try it first. Just don't ask me how I learned it.
GerryG