Changes made to Microsoft Family children accounts on https://account.microsoft.com/family are not applied to Windows 10 PC.
After logging in to the PC with a parent's account I found following message in Settings > Accounts > Family and other users.
We can't connect to Microsoft family at the moment, so your family on this device might not be up to date.
After checking that Internet connection was working, I went to Settings > Accounts > Your email and accounts, where following message was displayed:
You need to verify your identity.
After clicking Verify and entering Microsoft account password, both Verify message, and error We can't connect to Microsoft family... went away.
Note: in some cases instead of asking to enter your password, Microsoft may send a verification code to your email address.
Next step was to login with a child's account, where under Settings > Accounts > Your email and accounts the same Verify message was displayed. After verifying child's account the issue was resolved and Microsoft Family settings were finally applied correctly.