According to Jenny Jochens, and her Women in Old Norse Society the fate of the children of slave women was entirely at the whim of their owners. According to Jochens, the decision what to do with all new children in a Norse household rested with the male head of the family.

Whether the child was the offspring of a man and his wife, slaves, a man and his slave, or whoever, the decision for what to do with the child rested in his hands. Nor should you think this was an idle power. At their whim children, sometimes even legitimate children, were left to be exposed to the elements and die or disposed of by other means if the head of the household did not want to take them in. Laws against infanticide were only brought about because of the eventual conversion to Christianity and while the lives of slaves were still cheap and not entirely protected this did result in an eventual crackdown on the blatant disregard for the lives of infants, somewhat... In other parts of the Norse world, Iceland for instance, infant mortality remained extremely high even after conversion, due to the poor diets of Icelanders and the harsh conditions of the land.

Now even assuming that the child was not murdered out of hand their life as an offspring of slaves would not be an easy one, and it is likely, indeed all but certain, that they would share in the life of a slave as their parent(s) did. In some cases, perhaps if the child was sired by a free man, it is possible they would escape their condition but I am not personally aware of any examples of this, perhaps an expert on saga literature can chime in on the topic. But, once a slave the life of the child was at the mercy of their owners, and the lives of slaves in the Norse world was not a pleasant one.