You could telephone German Mobile Technical Support and it would probably be both easier and quicker for you to get the problem solved that way.
Click the following link and select Deutschland from the country list.
If the problem is solved could you come back here and tell us please?
the google translation is good enough. let me confirm the problems that you are running into, and i will have some follow up questions to better understand the issue that you are running into.
1.) you have two devices that your kids are using: identical moto E running android 5.1.1
2.) you, as a parent, are using a Note 4 - what Android OS version are you running?
3.) so you've turned on our screen time limits, and your child's policy is to allow device access at 5am, but you are still seeing it blocked at 9:30am, but how long it gets blocked for is different everyday (let's call this phone - Phone A)
4.) the second moto E - when the translation says "folds" does that mean it crashes, or the feature is working properly on the phone? (let's call this phone - Phone B)
5.) your note 4 is taking time to get the updated application activity from the 2 moto Es
1.) are the clocks identical on both Moto E devices, and are they getting their time updated from the network?
2.) "The app usage times are not recorded properly time in two phones, only VERY rough time recording." -> can you give an example of what you expect to see here?
if you can add screenshots that will help.
Marking this thread as Assumed Answered, as it is over a year old. Locking it.