@Ondřej-Šutera After some recent updates the behavior improved, indeed sometimes there is a notification showing up and the widget updates.
The main problem still remains for me that it does not update reliably. Sometimes it stops updating and I keep seeing the old forecast/temperature without knowing it, which is very bad. I would prefer to not rely on automatic updating at all if I could manually update each time, but the manual update doesn't work!
I think the biggest problem is that the manual update doesn't work. I can accept that the automatic updates can be buggy and not work on various platforms because of battery optimization, blocking of notifications etc. But why is it so difficult to make the manual update reliably work?