Currently, both Android + iOS use pull notifications, not push.
Unlike the Android app, there are no notifications at all on iOS if the app is closed though. (Only if in the background)
Push notifications are being added to Android right now (I have literally been working on it all week - a necessity as the previous method fails on Android 8+ anyway), and will appear on iOS soon after. It has always been on the road-map, just not implemented until now.
OK, so i timed it. I've waited now 10 mins, app in the background. And no notification bubble and when I go into the app, no games on the my games tab/page. if I hit the website here, it does show games having moves.
I waited this one out, and it's about 60 minutes until I get the badge notification and the my games tab is showing me games.
I did re-enable email notifications, didn't seem to make a difference, though I said I'd try it....and I did time myself for the 5 minutes to ensure I didn't go back into the app straight away. I do have the app set to a 5 minute refresh interval.