I have heard this word pronounced somewhat similar to [the River] Nile-ism as well as similar to Neal-ism. The former is obviously because of the German or possibly Russian pronunciation, but how should it be pronounced in English?
I'm afraid I vary my pronunciation of "nihilist / nihilism" to suit the sentence and my audience. "Nee-(h)il-ist", with a very soft "h", feels most correct to me, but for people who haven't studied philosophy (or who forgot it as soon as they left college), "ny-il-ist" is easily recognized from "annihilation" (I have never heard anyone pronounce that word as "an-nee-(h)il-ation").
My favorite sighting (hearing?) of "nee-(h)il-ist" is in the (NSFW) song 88 Lines About 44 Women by The Nails; at about 4:12 he mentions #39:
Terri didn't give a s**t
She was just a nihilist
There are two exceptions to Latin pronunciation. (I should say, two main exceptions. There may be others.) "H" is silent except in two words: mihi and nihil, where it represents [k]. Of course, the medieval spellings 'michi' and 'nichil' are now considered incorrect. I have heard nihilism pronounced with a [g] or [k] and it's not incorrect.
Since the root "nihil" comes from the Latin, I pronounce the way the Catholic Church folks use(d) it. Many will remember the words "imprimatur" and "nihil obstat" in books like the Bible, et al. The first meant "let it be printed" and the second something like "nothing to be objected to." Anyway, the "nihil" was always pronounced nee-hill or maybe nih-hil, but not nye-hil. So I had heard the Latin-y pronunciation for years. That being said, I recall a Latin professor spending much time reminding us that no one knew for certain how Latin was pronounced, though clues might be inferred from Romance languages. He taught Cicero, but reminded us (often) that it could be Chee-cher-o, or even Kik-er-o. So, in brief, I'm staying with the nee-hilists and the nih-hilists.