My recent research yielded some results.
We've been updating our Watch app, which is WatchKit based, to support watchOS 10 features. We've added new SwiftUI scenes to it, and the app itself works like a charm - no issues whatsoever. However, it appears that WatchKit apps have issues with WidgetKit complications - they won't update (or will have a significant delay) even when the app is in the foreground (such updates do not count towards the daily update quota).
However, this issue is not observed if the Watch app is pure SwiftUI - widgets will update immediately, including Smart Stack widget. Therefore, we are rewriting our Watch app to SwiftUI and getting rid of ClockKit completely (doing this forced us to increase the minimum watchOS version to 9.0 🤷♂️).
Hope that will be helpful to someone. 🤓
P.S. All said above doesn't solve the issue with data communication not happening between iPhone and Watch if the Watch app is in the background.