MotherShipton I have tried the app, set the battery to unrestricted upon first run. Notifications are turned on by default. I created a test appointment, set reminder at the start of the event. Backed out of the app and did other things in the meantime. I was alerted at the time of appointment.
In the past I had problem with Tuta app not autostarting after reboot, this seems to have been resolved. You have to make sure you add reminder when creating event in order to get notified.