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.