Nose slowly rots away if you get a skin cancer there?

Early treatment!: Cancer cells gradually invade and replace normal tissue. When skin cancer is more aggressive and fast growing the tumor can outgrow its blood supply. This leads to death or necrosis of the part of the tumor that is deprived of oxygen and may appear as a crusty or ulcerated area that may have a foul odor. So it is important to have suspicious areas checked and to treat skin cancer early!

Get treated: Otherwise, this is an understatement. However, the treatment of most skin cancers on the face is simple, and the vast majority are amenable to cure. Today there are even topical medications that may be effective, though i would myself opt for surgery.

Yes: The skin cancer will spread and be locally destructive which can result in complete loss of the nose. It can then spread to affect the rest of the face and can even spread in the nerves to the brain. Some types of skin cancer can metastasize into the bloodstream or lymphatics and then spread to the rest of the body which can be fatal.