Snapdraogon 8 Gen 2 is likely to be released in November 2022. Qualcomm has announced that the next Snapdragon Tech Summit will be held from November 14 to November 17 on their official website. At that summit the company may release the next generation of flagship chipset Snapdragon 8 Gen 2.
Xiaomi usually uses the newest processor from Qualcomm compared to other OEMs like Samsung etc. With that being said Xiaomi 13 series could be launched after in a short time with the release of Snapdragon 8 Gen 2.
The announce of Snapdragon summit has been removed from the website later on. According a famous blogger on Chinese website says SM8550 platform manufacturing is on process for more than a month.
Xiaomi 12S series will be equipped with Snapdragon 8+ Gen 1 and in the best case scenario Xiaomi 13 series will use Snapdragon 8 Gen 2. The first rumors appeared as an unusual 1+2+2+3 CPU combo with one Cortex-X3 unit, two Cortex-A720, two A710, and three A510. (platform unit info via GSMArena)