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Although I can't be sure of the range of dates when all men seemed to wear braces to keep up their trousers, I can say that it was entirely normal while I was growing up. Even when younger men started wearing belts, older men continued to wear braces. My father was still wearing them until he died in the 1970s.
Braces consisted of elasticated straps that went over each shoulder and buttoned onto the insides of trousers. At that time, all trousers were sold with these buttons.
Braces came in various sizes and could be adjusted to suit the height of the wearer with toothed clips that can clearly be seen in the photo of the boxed braces.
I suppose that braces must have been much more comfortable than belts, particularly for men with large stomachs - not that that was common in the war years and the years of austerity afterwards.