If you have a bathtub, you have a place to dry clothes: Racks can be set right in the dry tub. Collapsible wall mounted racks that stretch out over the tub can be installed. Retractable lines that stretch from one wall to the other can be installed, and unhooked when not in use. A spring-tension shower curtain rod can be pressed against two walls traversing the middle of the tub lengthwise and then clothing can be put on hangers and hung from the rod.
Where to hang laundry is one of those problems that it is great to ask a 90 year old. They will remember the days when there were no driers. And they would probably be amused that we even have to ask!
(My dad is older than that and he tells me women used to hang their undergarments
inside of the sheets and pillow cases out on the clothesline. They were shy to have their underwear seen even when they were not in it!)
Oh, and with a little child, do ask for the hangers whenever you buy him clothing that is displayed on hangers in the store. Some stores are happy to give them to you. They are almost always plastic, which makes them good for hanging wet things. Even after he grows the little hangers will find purpose in hanging smaller items of your own wardrobe.