I do wish to reiterate that "salad oils" are not all recommended. Some...like peanut oil...although they won't hurt you, do not really dry, and a slightly sticky, oily instrument might be less than fun. I doubt that many recorders are are made from softwoods.. (here are the best types) Since the recorder is from Hohner, it is almost certainly Maple or, if you are lucky, Pear.

But it IS possible to make even an average, relatively plain piece of wood shine decently with a carefully done sanding, oiling and, if possible. high-speed buffing. (I do this regularly)