Green’s Dictionary of Slang

high nose n.

[high-nosed adj.]

(US) arrogance, snobbery.

[US]L. Hughes Laughing to Keep from Crying 68: She’s got a high nose.
[UK]R.L. Pike Mute Witness (1997) 69: I don’t suppose that you know that Mr High-Nose Chalmers of the District Attorney’s Office has checked on every hospital.