But the two OP examples are entirely different. Cosy/cozy is not formed from a noun, nosy/nosey is.
Many nouns can be turned into adjectives by adding a “y” — buzzy, dirty, soapy, flowery, frosty, and lots more. In some cases the consonant at the end has to be doubled: muddy, nutty, spotty, gappy.
Only when the noun ends in “e” is there a question of whether or not to drop that “e”. In a few cases (as with nosy and nosey), both alternatives are used, e.g. maty/matey, homy/homey, poky/pokey. But in most, the “e” is dropped: bony, shiny, spicy, icy, juicy.