I can't speak to what was happening in European, Canadian, Asian, or African schools but can assure you that World War II was being taught in American schools while the war was still happening. In effect, American teachers have always incorporated current events into their daily routines. There are journals and artifacts from the 1860s that speak to how teachers were talking about the Civil War while it was happening. I have a textbook in my collection about Word War I that was written and published while battles were still raging.

If your question is about how did schools teach World War II after it was done, that's a bit more complicated. American teachers have long been a profession of committees, and as such, are fond of getting together and talking about their practice. The "National Council of Social Studies Teachers" was formally founded in 1921, though the idea had been around for a few years. Their early conference proceedings speak to how adults were talking through how to best help their students understand "The Great War" and by World War II, had established and recommended protocols for talking about current events. They advocated bringing in guest speakers, using clips from the media, and doing what adults have been advised to do for generations - follow where children go when dealing with difficult topics, don't feel compelled to give them all of the big scary stuff all at once.

The possibility of an individual child encountering content about World War II right after the war was highly dependent on where the child went to school, their school's educational philosophy, and even an individual teacher's connection to the war. American education is highly localized (with a few exceptions) and it's difficult to speak to patterns. One of the exceptions, though, is what happened after Sputnik. Although American history textbooks included World War II events as early as 1946, there is typically a delay between significant events and their distilling into textbooks. That process for World War II was interrupted by the space race and a collective shift to focus on math and sciences. So, in effect, America's teachers may or may not have been teaching the war immediately after it ended and any teaching around it in the 1950s and 60s was mostly "Nazis bad, America good" barring exceptions for specialized courses at the secondary level.