After doing some reading and making some observations, I'm becoming more and more convinced that we could see KHIII as early as January 2016:
'Winter 2016' is pretty ambiguous - it could mean either January, February, or December 2016. However, if the release was December, I would find it more likely it would say 'Holiday 2016' as clarification. The calendar in the trailer could either be December or January 2016, depending on whether you start the week from Sunday or Monday.
Bill Farmer seemed pretty adamant on Twitter that the game would be released by the end of 2015.
KH2 was announced in 2003. There was a blackout of information released because the game was announced too early; in an interview in May 2005, Nomura said, "It's not that delayed development has caused a delay in the release of the game. Recently, a release period has come into sight, and so we've started to release information." The game was released 7 months later in Japan in December 2005. Does this situation sound familiar? A slew of new information in the coming months could mean a release date is in sight for less than a year away. We know that at least some voice acting has been completed for the game - trailers with voice acting generally indicate that a game is near completion.
D23 is hosting an entire event dedicated to KH3 in Tokyo in November. I'd imagine we'd get another trailer, information about new worlds, characters, etc. A release date announcement is also possible. The important thing to note is that the release dates for KH games have almost always been announced no earlier than 2-4 months before release. If Square do the same with KH3, then an announcement in November would mean a release no later than March 2016.
I'm probably reaching here, but just following the patterns that Square has followed for past KH games, it seems logical that a release may not be far away as we would think!