Both my kids have eczema since birth. It's usually from food allergy or chemicals.
For both of them the red bumps didn't come up immediately. Usually it was just dry patch of skin, not scaly looking but dry when you touched it.
Since my husband nor I dont't suffer from either problem, I assumed it was chemicals first and replaced the bath wash, shampoo and moisturiser since we used these daily. We noticed an improvement from the next day. So now I only use sensitive skin formulas. Though still use normal laundry washing powder. Even now 2 years later when they have bubble baths 2 days in a row my daughter will tell me her skins starts to feel itchy.
Last time we went pediatrician advised me to use Physiogel if the eczema wasn't too bad. So I now keep a tube at home but rarely need to use it since we switched to sensitive skin products.