The vowel sound in "nose" is the open-mid back unrounded vowel, represented by the symbol /o/. It is a rounded vowel produced with the tongue placed in the back of the mouth.

In RP, the o in coffee is a short open back vowel /ˈkɒf.i/ , whereas in General American speech one often hears either a long open back vowel /ˈkɑ.fi/ or a long mid back vowel /ˈkɔ.fi/

Wiktionary offers two audio fragments that illustrate this difference

stamp

The vowel sound in "wash" is the /ɑ/ sound, also known as the open back unrounded vowel.

The AW is a caret O vowel sound (or/aw) as in draw, ought, caught, and taut.

(In British English, OR often has the same sound as AW because there is no R sound. So the words caught and court sound the same, as do lore and law.)