I recently had a similar issue with my kids.
What happens in the background is that whenever you open screentime for your kids, it actually requests the information in real time, that is why there is a delay. Apple doesn’t seem to store anything related to the screentime on the cloud.
This is both a blessing and a curse for us parents. When there is an iOS version mismatch it can create random issues. When the parents device is on a newer version of iOS than the kids, there is no issue as it knows how to talk to the older version as well. However, if the kids device is on a newer version, it might not know how to talk to it. When it gets an unknown response, it displays nothingness. Then you close it and it sends the updated version of nothing to your kids phone.
All of the above is based on my experience as a developer and anecdotal evidence I will explain below.
This happened to me about a week or two ago. I was trying to unlock an app for my kid and it went crazy as randomly it has before. I decided that night to sort out what happened.
What I discovered was that my phone was on 16.2 and my kid was on 16.3
Looking into the release notes I discovered there were changes to the screentime back end. With the phones side by side I opened it up on my phone and still was being stupid. Reset everything on her phone and opened it on mine again…. Reset.
I then updated my phone to 16.3.1 (hers was still 16.3) and now everything was working as expected, I reset everything on my phone, her phone updated as expected, I changed it on her phone, my phone reflected the changes.
I updated her phone from 16.3 to 16.3.1 and everything still worked.
I am making a lot of educated guesses here. I tried to confirm by having a buddy of mine that was having the same issue check versions and he also had the problem of kids having newer versions of iOS. Updated and everything worked.
Hope this helps someone.