My mother has the S22 and she only has Android 12 at this moment, she hasn't got any notifications or anything for upgrading. At this time. Not alot of people have access to it yet
It's only available for unlocked devices and it's currently only in Beta testing Yes Android 13 is out, but currently on Google Pixel devices first. Samsung UI needs time to test its compatibility with Android 13 It will start rolling out in a couple of months or so Unfortunately it will start rolling out to newer phones first. So the S10 series phones probably won't get it until the new year