A woman who is about to give birth (contractions or breaking water) is considered as a case of a hole sheyesh bo sakana, a patient in a serious condition. It is permitted, therefore, to help her, assist her and do whatever is necessary for her to arrive to the Hospital. Still, the Rabbis indicated that since as opposed for example to a patient having a heart attack, the mother-to-be is not in a life threatening situation, the actions forbidden in Shabbat (=melakhot) should be performed in an irregular way ('al yede shinuy) or by the help of a non-Jew. Thus, rendering those melkahot to the category of Rabbinical as opposed to Biblical transgressions.
Likewise, if one has to choose between taking his own car or a taxi, it is better to take a taxi, even if one has to phone-call the taxi, because parking the car (turning off engine, lights, etc) might be an issue with one's own car (yalqut yosef Shabbat D, 297). An ambulance is also a better choice than your own car.
All these indications--doing things in a irregular way or calling a taxi instead of taking your own car, etc.-- are suspended if one knows or suspects that by doing them the mother or the baby might be in danger. In case of a life threatening situation Shabbat is cancelled (hutra), which means that it has to be considered as if it was a weekday.
It is also permitted to travel with the mother, even when an ambulance is driving her and, technically, there is no practical need for someone else. We learn this Halakha from an explicit Mishna that authorizes to do a forbidden activity on Shabbat (=lehadliq ner) just to calm down the mother at such a sensitive time (liyasheb et da'atah shel hayoledet).
Once one knows that his wife might be giving birth in Shabbat, a handbag with all the paperwork needed for the Hospital and all the basic things needed for Shabbat ( like food, a siddur, etc.) should be prepared before Shabbat. It is also recommended to contact a taxi company or a driver and pay him for his service before Shabbat.
Candle lighting in NYC: 5:42 pm
Shabbat ends in NYC: 6:40 pm