Is a woman in labour (having contractions) still obliged to pray?

Yes, as long as postnatal bleeding has not begun and she is conscious, prayer remains obligatory and is performed according to her ability: standing, sitting or lying down.
The Prophet ﷺ told a sick man: pray standing; if you cannot, pray sitting; if you cannot, pray lying down (Sahih al-Bukhari 1117). Amniotic fluid or blood that comes out before the baby is born is not counted as nifas according to the Shafi'i school; it is like istihadah, so the woman cleans herself as far as she can, performs wudu or tayammum if she cannot use water, and prays. If the pain is so severe that she cannot concentrate, she prays as best she can without guilt. Nifas begins after the baby is born, and from then prayer is no longer obligatory until she is pure. The husband and nurses can gently remind her of the times. See the question on prayer after childbirth.