Check in Settings > Apps, and see if Outlook is allowed to run in background for power and data. On my Samsung the default was that Outlook was restricted, and so it didn't do notifications.
It is allowed
I will abandon Outlook since I can't fix their programming, and switch it back to gmail which has working notifications.
I am using a win 10 motion security program called ContaCam on a PC, it watches for motion detection using a webcam. If it sees motion, an alert is triggered and it sends an email to outlook on the phone. But the phone never shows any alerts, so it makes the whole process somewhat useless.
The other issue is gmail can not display an animated gif caused again by it's programmers. But I can use a different email client like spark, or blue mail which does run the animation. The animation is a miniature video of the motion event. It allows you to see what happened without going to the IP address of the webserver to view the full screen video. So I have to put together things to make it function like something reasonable. I was running up against 2 issues, Outlook wont send notify events, and gmail wont play the gifs. Crazy world.
Another oddity, I logged into the outlook email server using Spark email, and it still wont push a notification. Maybe the issue is a lot deeper than android Outlook app, it might be the way outlook email servers push emails to android. I read a lot of complaints about this lack of notifications going back years. And also badge numbers can not show up right on the Outlook icon, I have experienced that too.
Tomorrow I plan to switch Contacam to send emails to gmail, and use either Spark or Blue Mail to display the animated gif. Don't know what else to do really.
I would also like for the phone to have a lingering notification, not one that disappears. I have seen some alerts persist in the list, and others just appear for a moment and then gone. Know anything about that?
All this is for my personal use at home. I had an extra webcam unused and thought about setting up this for the front door. It looks out over the front walk and the driveway too. I thought about Ring Camera, but thought it too limiting and costs a lot too.