Nintendo isn't part of the "next-gen console wars" because they're not playing the same game (no pun intended) as the other two.
Let me use an analogy to emphasise my point. Coke and Pepsi are in direct competition. Both also have to compete with Schweppes Lemonade, as they're all drinks, all sugary, all carbonated, etc. But nobody pits Pepsi or Coke against Schweppes Lemonade. Why not? Because while the fundamental purpose is the same, they go about it a different way. Two go for a cola experience, the third goes for a totally different flavour. While some people aren't going to like Lemonades, and some aren't going to like Colas, many people are likely to enjoy both, and thus might purchase the Schweppes Lemonade sometimes, and Coke (or perhaps Pepsi) at other times.
On the other hand, few people simultaneously will go for both Coke and Pepsi. This creates a situation in which the two drinks have to effectively "fight" for the same market. Schweppes Lemonade, on the other hand, is only really competing for dollars, and for choice, they don't have to try to convince people that they should choose Schweppes Lemonade over Coke or Pepsi.
Bringing this back to the actual topic, the Wii U is a different gaming experience to the PS4 and XBO. If you want Nintendo games, if you want creative games with some motion controls and/or touch controls that have the feel of a AAA game, etc, then you'll get a Wii U. If you're big on FPS, if you like certain other types of game, if you like a so-called "mature" feel, etc, then you have to choose between PS4 and XBO. It's this choice that drives the console war.
In the 6th gen, Nintendo played the same game as Sony and MS, with efforts to appeal to the same sort of gamer. Yes, they had some of their own broader-appeal franchises, etc, but they didn't really differentiate that much. As such, they were involved in the console war. In 4th gen, it was even more obvious, with Nintendo vs Sega being very closely matched - result was that the console war was very hard-fought. In 7th gen, on the other hand, Nintendo didn't participate. People spoke of "PSWii" and "Wii60", because it was basically a given that, if you wanted the full spectrum, you'd get a Wii and one of the other two. On the other hand, "PS360" was usually used as a way to refer to things released on both systems, and relatively few people bought both.