I don't believe that there was really that big of a push to serve in those years as there is now, I think it is unfortunate that so many young men go on missions as a matter of cultural expectations or because that what their leaders tell them they should do, I think we would produce far better missionaries if the young men only went if they felt called to serve not simply told to serve.
I don't know why President Monson did not serve a mission, but there may have been a very good reason. He just happens to be almost exactly the same age as my father, who was drafted into WWII right out of high school. He didn't even get to finish his senior year of high school. He and his two brothers were all drafted and served in that war. When they returned home from the war, their widowed mother could not afford to send all three of her boys on missions at the same time, so the sent the one who came home from the war with a word of wisdom problem. My father always felt like he missed out because he did not have the opportunity to serve a mission, but after he retired, he made up for it by serving a mission with my mother. Maybe President Monson found himself in a similar situation.
Yes it was because of the war, as a matter of fact there are several apostles in the same situation. Some members think serving a mission is some sort of "rule", I am not so sure about that.
I would say that it is a general rule to which there may be exceptions. President Kimball said that every worthy young man should serve a mission, and every young man who is not worthy should get worthy. That is the rule, but there may be exceptions, as in the case of my father, who didn't have the money to serve a mission.