Here is the guidance the U.K. Consular Services Department provided me:

“If you are travelling with a child (under 18) and are not the child’s parent, or may
appear not to be the parent (for example, if you have a different family name), we may
ask you a few questions to establish your relationship with the child. We will always do
this as quickly as possible and in a way which is sensitive to the interests of the child
and the adult involved.

We hope you appreciate the importance of the additional checks we carry out for
children travelling into the UK and understand the reasons for them should you feel
you have been unduly delayed.

We do not wish to delay your journey any longer than necessary. It may help you next
time you travel if you could carry evidence of your relationship with the child and/or the
reason why you are travelling with the child.

This evidence could include copies of:
• a birth or adoption certificate showing your relationship with the child
• divorce / marriage certificates if you are the parent but have a different
surname to the child
• a letter from the child’s parent/s giving authority for the child to travel with you
and providing contact details if you are not the parent.“