How do I get my mobile data to work again, after the new Android 5.1 update?

I am having an issue with mobile data no longer working. Mobile data *was* working the day after updating to Android 5.1 on my Nexus 5. But now (1 day later) it is not connecting to mobile data. It just has the cellular icon, with an exclamation point. I've tried toggling data, airplane mode, WiFi, rebooting, factory reset, and resetting the SIM card. The last thing I tried is to call my carrier and go through 2 hours of troubleshooting. They said the only thing left for them to do is send me a new SIM card. It makes no sense. Everything else works fine. I can make and receive calls. Texting works. And WiFi works as well. Just not the mobile data. The weird thing is that it stopped working on the first day of my mobile data cycle. But the carrier swears that all is well on their side.