Learn English – Yiddish loan words for the foolish & incompetent

I would say that my answer about when to italicize loaned words and phrases is also the most appropriate answer to this question:

I think this is a case where authors can decide for themselves where to draw the line. Or, if the writing is for a certain publication, the editors will have a policy for whether a given word should or should not [have diacritics].

(However, that answer got zero votes, so who knows? I still think it is a good answer.)

Various publications will have strict rules about what diacritics are used and when. For example, The New Yorker always uses diaeresis. Each publication seems to have its own rules for bolding, headers, section numbering, hyphenation, capitalization, reference citation, and so on. And there is no definitive format.

I believe that diacritics fall into this category, as evidenced by their generally inconsistent use (though internally consistent within many publications).

I don't think there are hard-and-fast rules when it comes to words that have been recently assimilated into English. The pronunciation that usually sticks is a halfway house between the original foreign pronunciation and what is easy for Anglophones to actually say.

To take your two examples:

A word like tortilla can just be pronounced as if it were an English word, tortia, without any Spanish accent. But you will still find English speakers who pronounce the double l, just like the do with paella and Marbella, and also the j of fajita and jalapeno as [dʒ], since they are unaware of the Spanish pronunciation.

The sounds in the Japanese word karaoke are a bit more foreign. The r in there is a very different liquid to the English r, English speakers aren't used to gliding from [a] to [o], and English words seldom end with with an [e]. Hence the slightly mangled /ˌkæriːˈoʊkiː/