How quickly can oral corticosteroids act on atopic eczema?

It depends: It depends somewhat on the dose of Prednisone used but it usually acts within a few days.

Within Days: For severe eczema and/or exacerbation of eczema you should start seeing improvement in skin within a few days. If you have a severe flare make sure you are given a burst with a prolonged taper as eczema tends to flare right back up if oral steroids are abruptly discontinued. Make sure you are following eczema treatment plan with daily moisturization, bleach baths, avoidance of triggers, etc.