Noticed that text message notifications stopped working when we upgraded our software from iOS 13.5.x to iOS 14.2. Tried 14.4.x, and today updated to 14.5.1. Still no notifications with text messages. Receive notifications from other APPs, just not the Apple Message App. This was happening on both the iPhone 11 now running iOS 14.5.1, and iPhone SE running iOS 14.4.2.
Upon closer inspection, I noticed a particular conversation on the Message list on the iPhone 11 was muted (a little night-time icon by the contact name) for whatever reason. Verified that Do Not Disturb was indeed turned off. Once I swiped-left and unmuted that conversation, started receiving notifications again on ALL conversations/contacts.
On the iPhone SE, no conversation in the message list appeared to be muted. So I picked one, swiped-left, and toggled it from Mute->Unmute. Now notifications started working again across all my contacts/conversations.
This is a software bug in iOS. Pure and simple. And this issue has been going on for 9 months now! Other App notifications on iPhone work. Only Apple's Messaging app doesn't. Over 300,000 reported incidents on the Apple Support Communities Forums. Calls to Support, they act like they've never heard of this issue. Reported this issue to Apple on their feedback form. I'm sure they've heard it before. And I hope it's fixed soon. Until then, try my workaround-hack. Hopefully it will work for you too...