I discovered this problem shortly after updating to iOS 14.3 about a month ago. I was at the grocery of all places, and noticed new message badges but no notifications. I work freelance, so getting timely notifications is very important.
Here's how I fixed it, and it's been a permanent fix:
I first went to Settings | Notifications | Messages, and found that Allow Notifications was inexplicably turned Off. So, fix #1 was simply to turn Allow Notifications back On. In my case, I also have Lock Screen, Notification Center, and Banners all switched On. Then, I went to Settings | Sounds and Haptics, where I found that the sounds for Text Tone were also turned off. Switch it back On by choosing a Text Tone.
Obviously, I didn't toggle any of these things off, the update caused it. I'm not sure how long it took me to notice it, either, but I don't think it was more than a day or two at most. Also, the iOS 14.3 update would not go OTA on my iPhone 11 with 128GB of storage, and plenty of room for the update. I had to do it through iTunes. I tried OTA multiple times, after reboots, clearing all of the open apps, etc., but nothing worked OTA.
I hope this is helpful. I was relieved that it was just a couple of settings to fix it, as I've read that many people are waiting on Apple to fix it in an upcoming iOS update. I've noticed that the current is 14.3, and I've been expecting a 14.3.1 release, but nothing yet.