My son's nanny gets cold sores... How likely is it that she could pass it to my son when she changes his diaper if she forgets to wash her hands?

None. You son will not get col sores in the areas the genital area and butt even if the area is touch without hands being washed . Cold sores are very specific in the area where the form. Cold sores as we know them form on the wet dry line of the lip.
Possible. There is a possibility that if she is not careful, with her hygiene tha the virus could be passed to your child.