After giving birth I realised that the skin around my fingers are always veri dry and flaking off, what should I do?

Diapers. It is possible that washing your hands frequently for diaper changes, or the diapers, is causing this. (i assume you are doing this now!) you can start with moisturizers like eucerin, wear gloves after you apply it at night, to see if that will help first. If not, then see you doctor to see about other treatments. And, congratulations of the new baby!
Washing hands a lot. Could it be that you're washing your hands a lot more since your baby was born? That could lead to dryness and peeling. If you're using baby wipes, they might have a drying effect also. A gentle cleanser, like unscented dove, cetaphil or others may hel;p. Also frequent use of a moisturizer, especially after your hands have been in water, like bathing, dishwashing, etc.