The story
From turn (verb) + up (preposition) + one’s + nose (noun), describing a gesture possibly universally understood as indicating contempt, disgust, or scorn which has existed at least since Ancient Greek times: see Ancient Greek ἐκμυκτηρίζω (ekmuktērízō, “to hold in derision”), from ἐκ- (ek-, prefix expressing completion or totality) and μυκτηρίζω (muktērízō, “to turn up the nose, sneer at”) (from μῠκτήρ (mŭktḗr, “nostril”)).
Etymology from Wiktionary, CC BY-SA 4.0.