I would like to know the difference between Final English Exam and English Final Exam.
"Final Exam" has a specific meaning as an exam which ends a course and tests all knowledge obtained in that course. It could mean either a final exam for an English course or (less likely, but also possible) a final exam which is written in the English language.
"Final English Exam" doesn't necessarily carry the same connotation, it is simply the last examination of your knowledge of the English language. "The final English exam before we begin the Literature section of this course".
They are probably the same in meaning. There is a possible subtle difference because "final" occuring before "english exam" suggests it is a last English Exam, or examination for an English class or course, while "final" before "exam" suggests it is the last exam, and it happens to be in the English language. But this is actually a rather contrived difference. If it were a class in an English-speaking school, and the language of the exam was to be English, this wouldn't even be worth mentioning.
If I were an editor of a newspaper, and one of my journalists submitted an article that contained "English Final Exam", I would probably change it to "Final English Exam", depending upon context.
The Google NGram viewer shows that "English final exam" makes its first appearance in 1971, but "final English exam" doesn't occur at all. See English Final Exam.
A final exam in any subject should be written "final [subject] exam". For example, for a mathematics class it would be better to write "final mathematics exam" rather than "mathematics final exam". "Final Physics Exam" is to be preferred to "Physics Final Exam".
In all these cases it is the Exam that is final, not the fact of its subject.