Being born alive was the first significant achievement for a medieval person. This event happened at home and with only women in attendance. Medieval doctors had nothing to do with birth, any more than your dentist will read a chest x-ray. It was only late in the period that university medical textbooks even mentioned the subject.
Newborn babies have soft bones and their limbs often appear curved. Medieval Europeans believed that the bones needed to grow straight, so they swaddled babies with legs and arms pulled straight. Babies spent most of their first six months wrapped into the same posture they’d assume again in death. Cradleboards were often used; babies could be hung on the wall, out of the way. This could be a much safer place for an infant to be, since semi-domesticated animals wandered in and out of homes. An infant left in an open cradle would need to be watched at every moment, and medieval parents who weren’t wealthy enough to employ nurses were generally also too busy to watch the baby so carefully.
Swaddling cloths were not likely to be made of cotton, since woven cotton came into clothing use only in the 12th century, as a luxury fabric in Italy. It was not until the 14th and 15th centuries that cotton was common enough to start to generate rags for the new paper industry, so it’s not likely that it was used for swaddling or sanitary diapering, either. Linen and wool were the everyday fabrics of the period; the well-to-do would have used linen toweling for their babies, while the poor would have used whatever cloth was available, probably worn-out fabric passed down from its original use as adult clothing.
The word “diaper,” in the Middle Ages, meant a repetitive geometric design painted on a wall. When weavers designed fabric to be used for towels, they used twill patterns that left threads floating across the surface to be more absorbent. Goose-eye twill, a popular towel pattern then and now, formed diamonds with central dots, very much like a diaper wall painting design. When wall painting became less common after the Middle Ages, the word “diaper” began to apply only to the goose-eye twill towels, and finally to the super-absorbent cotton twills used for sanitary swaddling of infants.