Notice how the high school graduation rate among 25-29 year-olds rose between 1940 and 1950. Those are people who graduated high school between 1929-1934 and 1939-1943. Before WWII.

This pre-WWII trend is consistent with the post-WWII trend, up until ~1978 when effectively "everyone" graduated high school. In other words, this has been happening since before WWII, and WWI had nowhere near the civilian mobilization as WWII, so the demobilized wives/children argument doesn't explain the inter-war increase.

As for college:

Notice how the number of 25-29 year-olds with college degrees rises dramatically between 1948 and 1951 and then slowly through about 1960. That's WWII vets using the GI bill to go to college (18-21 year-olds starting 1945 to 1948). Then the increase tapers off.

There's another significant rise between 1965 and 1978. That's boomers avoiding the Vietnam draft with college deferments. Then it levels off (actually, decreases) for 20 years until the late 90's.

The late 90's rise corresponds to the late-80's / early-90's push to throw student loans at anyone with a pulse.

Starting in 2008, the rate starts rising again: Young adults of all ages staying in or going back to finish college because the Great Recession killed a bunch of jobs. Plus, probably, some amount of people who started college around 2002 to avoid being drafted for Iraq (which didn't end up happening anyway).