My ds1 has had chronic eczema all his life. He has a daily bath with dermol / oilatum emollients. Then steroid creams and more moisturisers. Bathing stops his eczema getting infected, the emollients in the bathwater are a vital part of the hydration process.
As far as natural products are concerned - they don’t particularly help imo. It is best to avoid as many products as possible - definitely no soap, no bubble bath - not even “natural” ones. Plenty of moisturiser - we use cetraben and epaderm, both available on prescription. Oats can be soothing, but my son actually had an allergic flare up when we tried oats in the bath. We used aveeno cream for along time, which is oaty but he began to complain that it was making him more itchy.
Also consider your washing powder; definitely avoid fabric softener. We use surcare which is fragrance free non bio.
As for diet I am not convinced. We have never seen any diet related change in ds’s skin. I think a milk intolerance would have manifested itself before the age of 5.