Tingling of hands can be due to circulatory insufficiency. Cold water can cause vasoconstriction leading to less supply of blood in hands. Try to wash your hands in mild hot water and visit to doctor to know the reasons of this circulatory insufficiency.
No need to worry! It happens with some persons. Consult your doctor.
I'm not sure what cause the tingling, but it could be as simple as using commercial soaps that contain harsh ingredients. Personally, I use handcrafted soaps made with mild and gentle, natural, ingredients. This way, I can rest assured knowing I'm ridding germs and providing my hands with healthful and nourishing skin care.
Try using a different soap or other preparation and see if this still happens. Meaning it could be an allergy to whatever you are currently using.