aside, the only period when "Nipponese" was used widely was World War II
, when Japan was officially known as the Dai Nippon Teikoku
or "Great Nipponese Empire
," and always referred to itself as such.
"Nipponese" was chosen over "Nihonese" or "Japanese" because it sounded tougher (I am not making this up). Of course, that ended pretty quickly when American GI's got a hold of the word and transformed it into "Nip," the preferred derogatory slang for the Japanese during the war. After the war, the Japanese mysteriously stopped using "Nippon" in English: now you mostly find it in abbreviations like ANA, NTT, and NHK.
The official name of Japan today is Nippon Koku, "State of Nippon," and the word "Nippon" appears in romaji on its postage stamps. Nihon is the preferred Japanese term for referring to Japan without specifically referring to the government. Only the worst of Japanology's pedants continue to use "Nipponese" in conversation or writing, but they're out there.