Since there is no organization that officially defines what is an English word, there are no authoritative criteria. But the following conditions indicate that the word has become assimilated:
--The term is used without including the definition
--The term is used without quotation marks or italics
--The term is used with English grammatical inflections
--The term is used to form other English words, through compounding, affixes, etc.
For example, the word mensch, from Yiddish and ultimately from German, usually appears in English without definition, quotes, or italics. You say “we’re all mensches here,” not “menschen.” I don’t know of any English words formed from it. But based on three out of four, I’d say it’s an “English word” now.
It is not uncommon for words to be loaned, altered, and then loaned back to the original language. A good English example is anime, which was originally the English animation, borrowed into Japanese and altered, and then returned to English with the sense of the particular style of animation that is popular in Japan.