Microsoft are just rolling out some changes – and they should be hitting all Project Online tenant sites by now – which allow for Project notifications and reminders to be sent to accounts outside of Exchange Online.
As long as you have a ‘Work email’ configured with the desired address for the user they can get alerts for new task assignments and various levels of reminders. There is a setting that governs notifications working even if you do have Exchange Online – and this would need turning on too – so I’ll start with there.
Go to PWA Settings, Additional Server Settings and scroll right down to the bottom – and the Turn on notifications option will be found. Check the box if it isn’t already checked.
Full blog post here: