They hung the washing on a line in the garden. Well, that's what my mother did, and she wasn't bothered if it rained as she said it made the clothes smell nice.
If it was winter and the temperature dropped, the clothes froze on the line. I remember that, but I have no recollection of clothes being dried inside because, like you, we had no central heating.
She told me she never used to go to school on Mondays when she was a child because it was washday, and she had to help her mother as it was a real chore that took all day.
As a kid in the 60's I recall being in the cellar at my grandparents home where the washing took place, and from memory there was a large tub with a gas ring underneath to boil the water, another large tub to rinse the clothes in, and a hand powered mangle to squeeze the water out. Packets of Robin starch were in abundance to stiffen up the collars of the shirts, and a clothes line was outside in the back garden.