Really sorry you're experiencing this.
We went privately at Great Ormond St for my newborn as drs adopted a wait and see attitude. Her eczema started patchy but rapidly was covered head to toe, in creases, raw etc.
We were given the following advice:
Steroids initially (thin layer) once per day in the morning after a bath (see below) but to be weaned off once under control. We received Daktacort for face, nappy area and creases, with Eumovate for everywhere else.
5-10 min Baths twice per day - not too hot or cold water with with QV oil. Initially i thought I misheard but figured the consuktant knew better.
QV cream - After the baths, at nappy changes etc.
Where breakout is v bad buy baby body stockings (really weird looking things) slather up baby with cream, pop on overnight to make sure the cream is absorbed.
It took a while, but touch wood we rarely need the steroids now. I've found moisturising with the QV cream obsessively is key.
We did allergy testing as consuktant queried a milk allergy but touch wood she was negative.
Obviously the above was prescribed for us, so will not necessarily work for you. Really hope you get things sorted soon.