For a good replica of the nose, here's an option:
Sculpt a nose out of
Sculpey and bake it according to the directions. Since the Sculpey nose will more than likely be too heavy and/or too awkward to attach to the puppet, use
Plasti Dip. Dip the nose in the Plasti Dip and build up a fairly thick coat around the Sculpey nose. Then slice it open from the back and carefully peel it off the Sculpey. I'd mix the blue and the black Plasti Dip to achieve the navy color of the nose. If you want the nose to be hard, brush some wood glue or epoxy on the inside (maybe several coats). if you want it to be soft, stuff it with some foam or fiberfill. Glue it to the head and your done.