Poor children - there are some very sad stories here. My DS (now 3) had bad eczema as a baby (starting from about 2m), and scratched himself bloody on his wrists and ankles (scratching his ankles with his toenails!), causing infections. He has also been admitted to hospital for bronchiolitis, which last year we had diagnosed as tending toward asthma.
His skin got better last year (2-ish) as we cut his baths to once a week (baths are incredibly drying, even with all the Oilatum crap) and started using Aveeno (oat based - really nice stuff). They seem to do quite frequent 1/3 off deals in sainsbury's, so we stock up whenever we see it, and now could bathe in it ourselves, we have that much!
HOWEVER, I really want to stress that some things seem to work for some children, and irritate others. It seems quite evident that eczema is a symptom, not a unified condition caused by one thing, and so please don't be discouraged if something someone swears by does not work for you. Keep trying different things, and remember that the most important thing with eczema is for the skin to be moisturised - however you do that, with whatever doesn't irritate the skin.
Although the eczema was getting better, we are still concerned about asthma, so managed to get an allergy test (not in the UK - the referral never came through), and milk products showed up. We have cut out dairy for 6 months, and having seen the specialist again, are looking at reintroducing it again soon - fingers crossed.
My former boss has a daughter who has had a terrible combination of allergies, and only outgrew her eczema at around 10yo. That is a long wait, but should give everyone hope for longer!
Good luck to everyone going through this! 