Reactions to washing detergents are very rare, even in those with sensitive skin and eczema. Studies of people with allergic contact dermatitis show no reaction to laundry products in nearly 100% of cases. 
If your baby wears clothes washed in normal detergent without issue, their nappies can be washed in the same detergent. If an all over body skin reaction occurs ensure you get a diagnosis for skin conditions and comprehensive allergy testing by a medical practitioner or specialist. If nappy rash occurs visit the GP for diagnosis.
For more information on what is in detergents and the function of the raw materials see the Mainstream Detergents, Fragrance in Detergent, Enzymes, Optical Brighteners, Plant based, Sensitive & ‘Eco’ Detergents pages and The Wash Wise Detergent Ingredients Info Sheet.