Thank you for your question.

As you likely know, the sugya of Milchemes Mitzvah is broad and complex, with different approaches in the Rishonim. I will focus on your specific point, whether there can be, at least theoretically, a Milchemes Mitzvah without a Jewish government.

The Rambam (Hilchos Melachim 5:1) writes that a war of "Ezras Yisrael mi'Yad Tzar" is considered a Milchemes Mitzvah. Many understand that whenever the war is aimed at saving Jews from an enemy, especially in a defensive context, it falls under this category. According to this, it would seem that Milchemes Mitzvah can apply even without a Jewish government, since the defining factor is the nature of the war itself. See also other questions on the Sugya you referenced on our site for additional applications of this idea.

That said, other Rishonim may require a Melech and Sanhedrin as part of the definition, as appears from the Sefer ha'Chinuch, Mitzvah 425.

With this background, we can address your question regarding World War II. The classification of a war as a Milchemes Mitzvah has broader, more communal implications, such as whether even a Chasan must join, and other Nafka Mina's, including the extent of one's obligation to enter a situation of possible danger to save another. Under normal circumstances, one is not obligated to do so unless it is a clear Milchemes Mitzvah.

There is no doubt that a Jewish soldier drafted into the Allied armies who fought against Hitler YM"S and his allies fulfilled great and significant Mitzvos, rooted in Piku'ach Nefesh. While one might suggest that such fighting could be viewed, in a limited sense, as a personal Milchemes Mitzvah, the simpler understanding is that Milchemes Mitzvah refers to a war conducted by a Jewish army or a defined group of Jews. Therefore, it is difficult to classify World War II, in its broader framework, as a formal Milchemes Mitzvah, even though the individual acts themselves were certainly of the highest value.

Kol Tuv,

Aharon Steiner