Über is seemingly a much smarter way than traditional car service to get to the hospital on Shabbos. If ones wife is in their ninth month, and they might need to go in on Shabbos, setting up The Über app before Shabbos seems to save a lot of Chillel Shabbos. (No phone calls, no paying.) So I called a Rov on Friday (Rabbi Kraus of Lakewood's Bais Horaah) and after telling me that he has to think about it, he agreed that it is the case (ask your own Rov though). He told me to practice pushing buttons with my nose, so that on Shabbos I could summon them with a Shinui (if possible).
If you're using this, make sure to shut off screen lock and plug in your phone so it shouldn't die. Make sure you have a valid credit card on file. Type in your address and the destination hospital address. Click the arrow to the next screen and set the phone somewhere that you'll be able to click the summon button with your nose. You should probably have a car service number handy just in case there's no cars around. I had this running over Shabbos and it used around 100 mb of cellular data.
I apologize for making a separate thread for this. I only did so because it's a Shabbos question that is potentially relevant to almost everyone.