So the list of due task correctly shows up in the today widget, Notebooks’ icon shows the correct number of due tasks, but no alert is played at the alert time. What is wrong?

First of all, the alert time needs to be in the future, otherwise Notebooks won’t play an alert.
Next, you need to assign an alert time to a task; a due date alone does not imply an alert.
Finally, you need to allow Notebooks to play alert sounds. You do that in iOS Settings > Privacy.

I Don’t Get Notifications for Due Tasks