nitramluap
I think I figured out the problem (in my case at least).
So, initial state:
iOS 14 and MacOS BigSur:
Both for Development (installing app from Xcode 12.5.1 to device) and Production (installing app from TestFlight) synchronization works very well. Quick and accurately.
iOS 15:
For Development environment (installing app from Xcode 13 RC to device) synchronization work ALSO well.
BUT for Production environment (installing app from TestFlight, app was submitted by both - previously Xcode 12.5.1, and now Xcode 13 RC) synchronization FAILS. It works just BEFORE first change of data on iOS 15 device. At first data changes on iOS 14 devices and MacOS BigSur are available on iOS 15 device, but then after first change of data on iOS 15 device it becomes out of sync - no changes are going to iCloud and no changes are coming from iCloud.
At the same time synchronization for iOS 14 devices and MacOS BigSur keeps going on without issues...
Because there was difference in Development and Production environments behavior, I've looked at my Schemas in CloudKit dashboard. There was a new field in Development Schema - CD_image_ckAsset, that wasn't there before (and I didn't do anything to add this field). I've Deployed changes in Schema in Production.
After that my TestFlight app becomes alive in terms of synchronization on iOS 15 devices.
Try to pay attention to Schema in CloudKit dashboard. If in your case you can't go even with Development environment, there also might be problems with fields in Schema.