Hi Justin_900,
Yes, sorry that was a typo. My phone is running iOS 15.4.1, not 15.5.1. The problem occurred since I setup my phone as new and has persisted through the 15.4.1 OS update (it's a new SE3 and came with iOS 15.4.0, I think).
When I setup the phone initially everything sync'd with iCloud and my reminders are present in the Reminders app on the phone, but I'm not getting notifications for them (I do on my Mac). The first step in debugging was to try to go to "Settings->Notifications->Reminders" and ensure that notifications were turned on. The problem is "Reminders" doesn't appear in the list under "Notifications" in the settings. I can't turn them on, or off, as the app doesn't seem to exist as far as Notifications are concerned.
I've then disabled iCloud for Reminders, deleted the app, re-booted the phone and then reinstalled the app. After re-enabling Reminders in iCloud again everything syncs up, and appears to work normally in the app, but as far as Notifications is concerned it's as if the Reminders app doesn't exist and I still don't get notifications for them.