Newborn baby wishes generally revolve around congratulating and encouraging the couple, so a simple "congratulations" will work. Words of encouragement are also acceptable.
As long as the words of encouragement support the couple's new addition to the family, they are considered newborn baby wishes. The following phrases are examples of newborn wishes: "You two deserve every bit of happiness this baby will bring you," "Warmest congratulations on the birth of you new baby girl/boy," "Can't wait to see that little smile," "I can't wait to help with the baby, if you need anything just text me" and "Love just got real. So excited for you."