What should I do if my child's Apple Account password reset fails?

This support community is volunteers sitting at home who want to help their fellow Apple product users out of the kindness of their hearts. We are just like you, not official Apple support. Of course then we have to decide if we can actually help a user. Some questions I look at, then shrug my shoulders and go onto another one. Many of those are ones where the person really has provided inadequate information for me to do anything other than start a game of 20 questions, which I don't have time for. Others I simply have no idea what the solution might be. Some issues can only be addressed by Apple itself.

Frankly I don't know of a single reason for seeing a pinwheel. It could be the device has stalled, or it could be it cannot connect to Apple. Try restarting the device. Make sure it can connect to the Internet and that everything between you and Apple's server for doing this works. Do you have a good wifi connection? I don't even know on what equipment you are trying to do this. Have you tried using a browser? Use this link to check the status of Apple’s various services and servers —>

Apple - Support - System Status

If it is Apple you want, Refer to this page for Apple Support features ➔ Contact - Official Apple Support

Select from the presented options until you find a solution for your issue, or see if there is a chat or phone call contact method offered lower on the page (you sometimes have to narrow down the options multiple times before this is shown). If you do not see your issue, keep experimenting with any series of selections until you reach one that offers a chat session or a telephone call and get the representative to redirect you.

This support article has various country telephone numbers for contacting Apple for support and service ➔ Contact Apple for support and service - Apple Support

Don't transfer the account to a different phone or else you will likely be back here with a new set of issues. To truly transfer an account to a different phone you need to completely reset the phone and then do that again when you transfer it off. If you want to try a different device, use a web browser instead.

Also make sure any VPN is disabled.

Make sure the phone can connect to the Internet. Sometimes wifi and/or cellular services can go down. These those using other services that may need them such as an app.

Oh, and remember that while there may be a billion product users, in reality I am thinking there's only a 20 or so people in the entire world that currently contribute 90% of the responses on this particular forum. Then you posted in English so that means your help base is going to be in an English-speaking country, so think most N. America and the UK (there are but a few Australians here). You posted when it was evening in the UK so I suspect most were busy with home evening stuff. Even for the USA, I know I had a lot of other things going on at home yesterday afternoon. ;-)