How commonly can lung cancer spread to the nose?

Extremely rare: Extremely rare. Usually late stage lung cancer metastasizes to bones, liver and brain.

Exceedingly rare: I've never heard of lung cancer metastasizing to the nose. I can't say that it has never happened, but the most common sites of spread are the bones, brain, lymph nodes, and surrounding tissues. The nasopharynx is not even a consideration. A second primary cancer can develop there at the same time as a lung cancer, though.