I live and work in Japan as a translator and am fully bilingual, with a number of bilingual camera geeks in my social circle (some of whom actually work for Nikon).
I can confirm that there is no "correct" way to pronounce the word. Bilinguals when speaking Japanese pronounce it ニコン, which cannot be correctly replicated with English phonemes (the i is like an "ee" sound but shorter, the o is an "oh" sound but shorter, and the final n sound just doesn't exist in English). When speaking English, we pronounce it as Nigh-kon when speaking North American English and Nick-on when speaking British English.
In conversations with North American bilinguals, many voice the opinion that it sounds pretentious to pronounce it Nick rather than Nike when speaking with a North American accent. But that's a matter of personal taste, not correctness.