Both Oxford English Dictionary and Merriam-Webster agree that "nitch" was the correct pronunciation in both British and American English until very recently. You already linked Merriam-Webster, so here's O.E.D:

N.E.D. (1907) gives only the pronunciation (nitʃ) /nɪtʃ/ and the pronunciation /niːʃ/ is apparently not recorded before this date. H. Michaelis & D. Jones Phonetic Dict. Eng. Lang. (1913), and all editions of D. Jones Eng. Pronouncing Dict. up to and including the fourteenth edition (1977) give /nɪtʃ/ as the typical pronunciation and /niːʃ/ as an alternative pronunciation. The fifteenth edition (1991) gives /niːʃ/ in British English and /nɪtʃ/ in U.S. English.

(N.E.D is the original name of the O.E.D. "/nɪtʃ/" is pronounced "nitch" and /niːʃ/ is pronounced "neesh".)

I hope cache isn't pronounced like cachet (rhymes with sashay) rather than cash.

Well, crap. Guess I'll eat a nice quitch to chear myself up