What tool is needed to adjust this headset on a children's bike?
The bottom ring is used to provide tension against the bearings to stop the headset from wobbling around. It only needs to be hand tight, hence no spanner/tool attachments.

The middle and top ring are tightened against each other, as a lock ring to prevent the bottom ring from loosening.

The top ring can be tightened with any spanner of the right size.

The middle ring requires a specialised spanner called a C-Spanner, a hook wrench or a hook spanner. It looks like the image below and is readily available from any bike shop or online store. Eg: Chain reaction sells one from Park tools