do immigrants need TOEFL/IELTS/TSE?

The big issue is where you are now? Do you already have a green card? If you have one, then you will only need to take the NCLEX-RN. The language requirements are only if you need to obtain a green card, and are part of the requirement for the Visa Screen Certificate, not for licensure.

Moved this thread to the International Forum, since it deals with issues not related to American RNs working in CA.

Local exam is not required at all, and it never too late to back out. The next exam will not be for months there.