HI my daughter is now 4 months old and has been recently diagnosed with eczema. However she had the same problem starting at about 2 weeks old, and I was advised also by my health visitor doctors etc just to leave it. It progressively got worse upto about 8 weeks, where the advise was still the same, by this time it had spread all over her face and scalp area.
But by then i decided to do something about, being at a advantage of being a pharmacist myself, so I should have followed my own advice earlier I found the following helped manage the condition and eventually clear it for a few weeks maybe had I started doing this earlier my baby wouldnt have been sufferring from eczema today.
things I did from birth to minimise irritation
Avoid use of any products, I used only water for bathing, nappy change etc
olive oil bp for moisturising the body after bath or massage
additional things introduced at 8 weeks for me which helped but fine to do at 2 weeks:
bathing area with rash with cooled boiled water and a cotton wool and applying vaseline-this can be done as often as required througout the day
the key is to keep the skin from dehydrating and drying out, therefore do this many times througout the day, and if it persists upto baby being 8 weeks at this stage baby may need a prescribed emmollient /bath additive e.g doublebase
hope this helps