wow meep, the exact same thing happened to me when my lo was 4 months, I've always had eczema but it went crazy, my eyelids swelled up and I had eczema just all over my face and neck.
I went to the doctors and he sent me away with an enormous carrier bag of stuff. I had antihistamines (3 a day, piriton) betnovate from my body, and betnovate RD (mild strength) for my face/eyelids, aveeno bath oil and body lotion. I bought the aveeno face cream myself from boots, it's good for general use but on bad eczema can sting a bit.
To be honest if it's a big flare up, it can take weeks to go down with just diprobase and steroid creams like betnovate can get rid of it, although they should be used very sparingly before applying an emmollient like diprobase/aveeno/doublebase etc
I definitely wouldn't exfoliate the flakes off at ALL, but especially not with an exfoliating facewash, they're asking for trouble! The flakes are horrible but scrubbing them off can make the patch of eczema spread outwards.
During a flareup on the face I use emulsifying ointment melted in hot water, and a soft cloth or cotton wool, the emulsifying ointment softens the skin, then I pat dry and immediately apply betnovate RD then a couple of layers of aveeno cream.