I suffer with Excema, particularly on my hands and neck. I don't train indoors, only ride my bike outdoors which means I get wet in winter and hot and sweaty in summer.
My hands had just started to clear up last week when I did 100K on Saturday in my neoprene winter gloves which turned out to be overkill and I ended up with hot sweaty hands all the way round which have now brought the excema back.
I'm afraid you may not want to hear this but it's something I have learnt to tolerate rather than try to cure completely. The advice above is not bad regarding dairy and also consider how much wheat you eat. I find showering in natural products rather than soaps to be better. SANEX no soap is evil and proved to make my symptoms much worse. I use Body Shop olive oil shower gel. As has been said above use rubber gloves when washing up and also when you wash the bike and car.
Interestingly, my doctor told me that Aqueous cream should be avoided, then prescribed me with Cetroben cream for use twice a day (at least). I compared the ingredients and although they don't have quantities by percent, the active ingredients are practically the same. I suspect pharmecuetical persuasion at work here!
When it's particularly bad I try to alternate between a gel called Eumovate and cream called Fucibet (both prescription). It seems to help but doesn't ever fully go away.
I saw a specialist a few years ago who said I would just have to live with it and treat it. I think in general the NHS aren't interested and I haven't found the time to go and find an alternative.
Sorry, in reading your OP I suspect it's not what you wanted to read but this is my experience.