It actually wasn't even until the 19th century that they were called diapers. From at least the 15th century onwards they were referred to as clouts. The term diaper actually comes from the diaper cloth that was commonly used for them. It's a particular type of weave, usually in linen, that creates a very absorbent cloth. Other items were made out of it as well, such as waistcoats. Over the clout you could have a pilch made of a wool which would help prevent leaks, but wouldn't need to go through the rigorous cleaning as often.

Though washing clothing could certainly be more difficult in winter, clouts would often be scoured and washed in boiling water. Hot water was something they would certainly have had going near the fire at almost any time. Hang them near the fire to dry and laundry does not need to leave the hearth area at all. Undergarments were washed in the same way, allowing people to stay clean during any season.

While fabrics could be expensive, there has always been a wide assortment available that included affordable qualities for anyone (simply, we can't have naked people). Diaper cloth would not have been expensive, but any rags could do if need be. Clothing gets remade many times in it's life in order to make use of fabric, eventually reaching a point that it's not good for much other than rags. Likely it saw through two or three garments before that point, so it's well returned on it's investment by then.