According to the Longman Pronunciation Dictionary it is:
niche niːʃ nɪtʃ ǁ nɪtʃ — Preference poll, British English: niːʃ 95%, nɪtʃ 5%. In American English only nɪtʃ.
niche niːʃ nɪtʃ ǁ nɪtʃ — Preference poll, British English: niːʃ 95%, nɪtʃ 5%. In American English only nɪtʃ.
However, I'm American, and I anecdotally disagree that "nitch" is the only pronunciation. I exclusively use "neesh" and I regularly hear both pronunciations from other Americans.