Is "adios" an English word now?

In essence you're asking whether or when "a common Spanish word that many [English speakers] happen to know" becomes an English loanword.

Loanwords are distinguished phonetically (they have sound combinations that are uncommon to English). They have a form similar to their origin language, but perhaps have modified spelling or (especially for English) optional diacritics. (Compare resume and the French word résumé.) Adios fits this pattern, coming from the Spanish adiós and being adapted to English vowel pronunciations. (Compare this video presenting variations in American speech with the Spanish version. Now, imagine adios in a deep Southern American drawl. It happens.)

The Oxford English Dictionary cites instances of an Anglicized "Adios" or "A dios" as early as 1592:

1592 T. Lodge Euphues Shadow sig. L With a friendly Adios he left him to his rest.

Its use within English syntax as a noun modified by an adjective in an ordinary way is significant. While its context remains Spanish, the word stands alone in English to denote a Spanish goodbye. It can also be used in English where other goodbye words would be expected, like this 20th century example:

1940 C. McCullers Heart is Lonely Hunter i. iv. 53 ‘Adios,’ Jake said. ‘I'll be back sometime soon.’

One could argue that Jake is code-switching between languages, but the general intelligibility of what he said among non-Spanish-speaking English speakers suggests that he's using a loanword. In closing, adios, aloha, and ciao!