I have eczema, acne, and sensitive skin, and DH had child-hood eczema and still has quite sensitive skin.
For washing liquid we usually use Fairy or Ariel non-bio, never had a problem with either but we make sure not to put too much liquid in. Make sure you follow the instructions on the bottle as to how much is the right amount to put in, if you put too much in you could irritate your skin. We also do not user fabric softener, partly because I don't see the point, but also partly because I've always had eczema flare-ups any time any was used on my clothing. At the moment we are actually using Ecover non-bio laundry liquid as well as their delicate laundry liquid for delicates. These are usually more expensive but we have recently managed to get them both on offer to make them work out similar to our usual brands.
I also got eczema from Cien stuff unfortunately although DH was ok with the mens things.
For my skin, cheap and simple products that don't irritate things are 'Simple', as well as Superdrug's own brand products (have to be a bit careful with some of the very fragrenced shower gels but otherwise mostly fine), and their 'B.' range of skincare.
We usually use Head & Shoulders shampoo & conditioner as it has zinc pyriothine in it which helps combat eczema on the scalp which I suffer from otherwise. Superdrug used to do a great cheaper anti-dandruff shampoo range but last time I shopped on their website it was not showing up so I think they may have sadly dropped the line which is a shame as it was fine and saving us money
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For anti-persperants I use areosols, Mitchum now have a totally unfragrenced one which I'm currently using and loving, I also like Vaseline's.
For lip balm, cheap and fine for sensitive skin is plain Vaseline or plain Nivea creme.
Sudocrem as a spot treatment for big cystic spots overnight.
Also if you're on a low-income and get free prescriptions, a lot of stuff you can get on prescription which could cut costs of things that are expensive to buy OTC for eczema e.g. Oilatum bath emollient, Oilatum shampoo, Diprobase emollient, E45 cream etc. I get most of that on prescription as I'm entitled to free ones and I couldn't really afford it if I had to pay OTC prices.