I am sharing the solution here for future reference.
We investigated the problem offline and it turns out the Widget component inside the BP referencing the Display Widget also had its Is Focusable property set to True. This took away the focus from the underlying referenced widget causing it to not receive the input unless the focus was manually restored with a click.
To make it work properly, the Display Widget should be the only one having its Is Focusable property set to True. All other widgets and widget components should have Is Focusable set to false.