Notifications can't be passed through in standalone mode- which occurs when the phone is disconnected from the watch. There's no way for them to communicate, for example when the phone is powered off.
You were probably in remote connection mode and didn't realize it. The phone is on a network (WiFi or carrier) but not necessarily in close proximity to the watch. The watch uses it's cellular connection to communicate with the phone.
When in standalone mode, there will be a watch icon at the top of the screen that is displayed on every watchface when it's active. This icon doesn't appear with RC or BT connection. The other way to check is to pull down at the top of an active watchface and it will display how it's connected.
I keep my phone at home, so I'm always in RC mode. It can be flaky. After the last update, it appeared to be more stable. But yesterday I noticed it was broken again. The solution is to do a soft reset on the watch. Yes, it sucks...
The only other option is to use the watch's number, and completely bypass the phone altogether. Then texts and calls will come to it directly,and thus work in standalone mode.