I don’t think it was anything related to a particular user’s behavior/setup on that list, for example, earlier in the morning when I attempted the same on my own device (pushing an order to Square POS via Orders API), I saw the same 5 minute+ delay for it to finally appear that the merchant was seeing just today. Now orders are showing up within a few seconds as expected which leads me to believe it might be something on Square’s end was causing the issues.
In my particular case I had 100 or so orders in my Order Manager, had the Square app active as usual, but no push notifications enabled. But again, the behavior resolved on its own (at least on my setup) without any changes.