I would definitely ditch the fabric conditioner, and put an extra rinse on each time you wash. I have eczema and fairy non-bio still brings me out - you could try surcare?
Make sure all the clothing next to his skin is 100% cotton, and make sure he doesn't get too hot.
If he's scratching, make sure his fingernails are short and use scratchmitts/socks if possible.
All sounds very 'non-medical', but it really does make a difference with eczema.
Once you've done all the above, you can therefore rule out all external irritants except for two - the oilatum etc, and his milk.
If it's still not getting better, then go back to the docs for different cream - my dd1 was allergic to everything with petrochemicals and parabens in it so we use dermasalve. It's important to put the cream on often, eg 4 times a day.
If it still doesn't get better once you've tried different creams, then you could consider an allergy/intolerance to his milk, but if the eczema isn't too bad and he seems settled otherwise, I wouldn't worry about that right now!