Although I stopped using the mobile apps early in the decommissioning process, one of the most useful things was push notifications to my phone about new Inbox notifications, such as comments or new answers. With the complete decommissioning of the mobile app infrastructure coming, it would be good to replace that one aspect with something comparable. Since RSS feeds are quite ubiquitous, people could use them in different ways. I would add mine to an RSS-to-email service and get emails about new inbox notifications. Others could use services like feedly or mobile apps that give phone notifications for new items.
A good example of this is Reddit, which offers users private RSS feeds of things like their homepage, using a secret token that can be invalidated by the user if it becomes public.