Hi SquidgersMummy just wrote a long post to you and accidentally deleted it all - grmph. So here a less organised version.
Firstly I'd recommend you have your little one properly tested for dairy allergy - insist on a referral if you get the impression she reacts to it through your breastmilk (I've written a post earlier about my dds excema history in another thread - excema Terror by Sangarika about the difficulties accessing eczema experts). In the meantime you could cut out eggs (apart from dairy the most common food trigger of infant eczema). Soya and goats milk have proteins very similar to cows milk, and many children react to both if dairy intolerant (my small daughter does), so switch to coconut milk, rice milk or oat milk (you get those at any health food shop). Make sure it's calcium enriched; if not take a substitue - ask the health visitor. If she discourages you use that as an argument to get to see somebody at the allergy clinic, then you won't have to guess...
Secondly I'd also recommend to switch fully from detergent to soap, not only in personal hygiene products but also in your washing - this has made a massive difference to my daughters skin after she was diagnosed with an allergy to the normally healthy bacteria that live on her skin. She is now virtually eczema free again, despite that allergy. Detergents are chemically different from soaps, and very aggressive to the protective skin barrier, so prospective triggers are more likely to get through. The most common detergents are SLSs: Sodium Laureth Sulfate and Sodium Lauryl Sulfate, but there are many more. Anything that foames could contain detergent: washing powder, toothpaste, shampoo, washing up liquid etc.
I now make my own laundry powder (or laundry gloopy mass I should say) by grating two cups of pure olive soap (only ingredients are olive oil and salt) and mix it with one cup of borax substitute and one cup of washing soda. This is then cooked up with five cups of water until the soap disolves. When it cools down I mash it through with the potato masher and then use three tablespoons in each wash, using vinegar instead of softener. You'll have to use this for all your washing for it to work, as otherwise there will allways be leftovers of detergent in the machine.
It washes about as well as any non bio powder, so I use an oxygen based bleach if the clothes get a bit dull, and work the laundry soap directly into any bad stains before washing.
It sounds like hard work but it isn't, and all the nights that we now sleep peacefully and my daughters happy skin make it worth it by miles.
For washing my daughter - hair, hands and all - I use the same olive oil based soap, and rinse her hair with very diluted organic cider vinegar. This works so well that I have given up on oilatum - it helped the dryness, but also made her skin red. I now just do the wet pyjama approach: covering the inside of her pyjama generously in emulsifying ointment before I put it on (nice and warm from the heating). If I notice her skin getting dry during the week I follow the recommendation of one pediatritian: Use diprobase right before putting on clothes and pyjamas and put it on so thick that you can write your name in it. If on the day after she is still dry and it looks like it's getting worse I use cortisone once, in the evening. But with the soap now it's rare that we need to use any cream or cortisone at all.
Sorry about the novel but I remember how helpless one can feel with a tiny baby suffering, and I really believe these two things could make a big difference.