It only seems to be some Americans pronounce it nitch, so maybe regional there, though I couldn't guess where or which is most common. Here in the UK essentially always neesh.
It is from French after all.
Edit: American audiobooks on the other hand seem to go 100% with nitch for some unexplainable reason.
It is fairly unusual for American not to Anglicise pronunciations so it seems to stand as odd one out. Then again you kept the original "correct" pronunciation of herb - it was British English that changed for some reason.
> It is fairly unusual for American not to Anglicise pronunciations
Anecdotal, but I think this is only true for French. For example, I think U.S. Americans (even those with no Latin American ethnic background) are much more likely to pronounce Spanish loanwords in a way that's closer to the original than Brits are.
If my anecdotal belief is correct, I suppose it's largely because of the very close contact between the U.K. and France on the one hand, and the U.S. and Latin America on the other.
I believe BEng distinguishes "fillet" (with a final t) from "filet" (without), the former having been borrowed from French when it still had a /t/ at the end, and the latter being a later re-borrowing (usually in phrases like filet mignon). Something like "filet of fish sandwich" comes across as a humourously fancy name for a pretty ordinary meal.
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.
I can't say if that is the case here, but that is what I would suspect. As a british person I have usually heard it and used it as 'neesh'