don't forget that it will only work with iPhone 15 Pro and Max versions since it's a Pro/Max feature...Apple has said iOS web push isn't coming until 2023.
iOS xx.2 is always before the end of the year. xx.3 is usually before the end of the year (iOS 15 was an exception where it was in January).
I'd bet money we aren't going to see it in 16.3 either. My random guess is 16.5... iOS xx.5 typically is mid-spring, which would fit within Apple's "sometime in 2023" timeline, but still before Apple's next WWDC.