I have this issue on my XR and newly 11 as well. At first, I thought that it was XR bugs after updating to 13.1.1. I have contacted Apple Support several times so they set a phone call for me. After then, I reported this issue to the personal advisor from Sept - now. They asked me to send my screenshots, remotely checked my 'settings' screen to check whether I set them accurately, data analytics, screen recordings, tested email sendings when the screen's locked/unlocked, restored to factory settings and etc. I would like to say that I have done everything to get it fixed but nothing works so far!
The advisor has asked the engineer team if I can downgrade to the older version, but they wouldn't let me do that too, nor it is about the hardware issue because I am also having this issue on my i11. I checked both phones at iCare and iStudio and they could only restore the phones for me (no luck).
The notifications are working properly on my 6plus but not recent XR and i11. Obviously, it is iOS issue but i'm so upset that the advisor mentioned that they haven't had any case like this before so it is like I have to keep reporting them my issue and the engineering team ain't working at all! since 13.1.1, now they released 13.2.2 still this issue persists and it is worse on my i11.
The delayed notifications, no alert/vibration. Sometimes alerts without banner or banner without sound/vibration or just delayed for minutes - hour. Everything pops up randomly. It is not stable at all. It has been months already and they aren't working to fix this. This is really BAD!