So anyway, I reckon you can finish the newly exposed edge without refinishing the entire neck, but matching the color and sheen might be a difficult thing to do. If it's currently finished in lacquer, stripping and refinishing the whole thing is comparatively easy.
But if it's poly, forget it, get a new, raw neck; then cut to desired shape (or pay Warmoth to do it), and finish to taste.
Better yet, go with one of the lovely tropical hardwoods, no finish required.