So, first, an obligatory statement about the 10th Amendment: Since education isn't mentioned in the Constitution, it's up states which means it's hard to speak with any confidence about what's taught in American classrooms writ large. I wasn't able to locate an in-depth look at your question but I'm familiar with books and articles that have looked at the history of teaching the Cold War, the Viet Nam War, and Civil Rights movements which looked at state standards and in textbooks in states that engage in a textbook adoption process and confirm what Joseph Moreau puts forth in his book - like most things in public education, the history of war is subject to political winds, and steering is mostly done by white, Anglo-Saxon, Protestant cis-men and there is rarely a single narrative.

That said, it's worth noting that schools were incorporating histories of the war before the wars were over. During World War II, national teacher associations in partnerships with colleges and universities provided professional development on how to teach the "history" of the war - the events leading up to it as well as current events as they unfolded. In my library, I have a book a light-fingered relative got as a Junior in High School in Buffalo in 1919. Presented as a living history of the war, his teacher used it as a part of the curriculum. This was also in New York State where textbook decisions are up to the district or school. (As compared to Texas which adopts textbooks for the entire state.) The New York State American and World History exams in the mid-1940's included questions about the war written in the present tense. The 1950 exam essay asked students to write about the rise and characteristics of 20th Century dictatorships. History class has rarely been entirely focused on the past.