The transliteration was done by the passengers themselves, by the shipping company that transported the passengers, or by interpreters at Ellis Island.

While many of these immigrants wouldn't have been fluent in English, they still would have had access in their home countries or through the shipping company to someone bilingual who knew English as well. It's not like English was some incredibly inaccessible language to someone in Poland or Russia.

Is there even a consistent way to do it in a language? The Russian chess player Ian Nepomniachtchi is called Jan Nepomnjasjtsjij in Norwegian for instance. Must be so many ways to do it.

Depends. Some non-Latin scripts (eg Greek) have standardised transliteration schemes. Others (eg Arabic) have no clear standard, so the transliterations are wildly inconsistent.