In biology classes every professor always did the nitch one (in an evolution/ecologic context) and it was not until about ten years ago or so that I ever heard the neesh one used … on a radio show and it jarred me. Made me think of some pronouncing the town Des Plaines “correctly” according to the rules of the its language of origin but not correct for what the town is actually called. IOW pretentious twit who does not really know the word. Since then I think the neesh one has come to predominate and I have gotten used to it.
Apparently the use in an ecological sense (rather than recess in a wall) originated in America in a paper from 1917 by a Californian scientist named Joseph Grinell. As that Abbot and Costello bit illustrates, Americans at that time would say that with the nitch version. That pronunciation goes back further though, which is why Lewis and Clark in 1805 named a cove “that dismal little nitch.”
Thus its modern popularity as a word that mean a place in the real or metaphorical ecological system came out of American ecology and was pronounced by those who coined its modern usage as “nitch” and many of us who learned the word in that context learned it said that way. But as it spread out of that academic context its pronunciation morphed to congruent with its French origin more often, especially out of America.