Welcome to the forum, zalfiesimmer! The first question is -- do you have the expansion pack, Get to Work? If you do, then not being able to go with your sims is a serious bug.
But I'm guessing from your post that you have only recently acquired The Sims 4. In that case, although you do have an option to send your sim to the hospital, nothing much happens except what you've described. If you want to see labor, get a bassinet from buy mode and click on that to have the baby when you get the notification that your sim is in labor.