If you are on the site at the 'questions' page and someone pages you with a comment, the red indicator will light up immediately. Similarly, if you are on the 'questions' page the green indicator will light up as soon as you acquire a new badge.
If you are viewing a question that has been submitted to the review queue, does the close counter react in real-time? Or is a page refresh needed in order to see the current count? Same question but for questions that have entered the review queue for edit approval.