There is no set age for this. There used to be when I first started, but the various Sensei in the dojo realized that age wasn't the best factor in deciding this. It depends on the individual person and various factors such as ability, maturity, size, and social factors. Here are a few questions to ask yourself.
Do I constantly feel like I'm reviewing in the kids class? There is less repetition in the teen/adult classes because most of the students tend to practice at home more consistently so there is less need for review.
Am I ready for a slightly more serious dojo experience? While both classes are fun and exciting, the activities in the teen/adult classes will be slightly more advanced and will require deeper concentration.
Am I the biggest kid in class? If you're a foot taller than every single student in the kids' class, you may want to upgrade.
Are my friends ready to move on to the adult class too? Your karate friends will be very valuable throughout your life. Sticking together is important. If you are considering moving on to the teen/adult class you may want to discuss this with your close karate friends as well.
Does the different time work for me? It's funny to think of it this way, but one of the biggest difference between the two classes is the time. Does this work with your schedule? I had to delay switching classes for a year because of my football schedule. There's nothing wrong with staying in the kids class a little longer if it works better for you and your parents. We're just happy that you're still training.
That being said, there is one way to figure this out that may be even more valuable than these questions. Come down to the teen/adult class and try it out! You can make up a class there if you missed a class, or simply come down for a class or two just to see if you're ready.
See you there!