Washing just the top and bottom of a newborn is a common practice for new parents. They do this because it's easier to clean the baby's bottom without having to deal with the umbilical cord.
The umbilical cord eventually dries up and shrivel in size. This is a natural process that happens after about six weeks of age.
Hold your baby on your knee or lay them on a changing mat. Take off all their clothes, apart from their vest and nappy, and wrap them in a towel, it's important to keep them warm.
Dip the cotton wool in the water (make sure it doesn't get too wet) and wipe gently around your baby's eyes from the nose outward, using a fresh piece of cotton wool for each eye. This is so that you don't transfer any stickiness or infection from one eye to another.
Use a fresh piece of cotton wool to clean around your baby's ears, but not inside them. Never use cotton buds to clean inside your baby's ears. Wash the rest of your baby's face, neck and hands in the same way and dry them gently with the towel.
Take off the nappy and wash your baby's bottom and genital area with fresh cotton wool and warm water. Dry very carefully, including between the skin folds, and put on a clean nappy.
It will help your baby to relax if you keep talking while you wash them. The more they hear your voice, the more they'll get used to listening to you and start to understand what you're saying.