I believe you should report the passport as missing and apply for a new one. It is a lot of hassle, I know. My (adult) son's US passport had expired so he flew from US to UK on his British one and then had to go to the Embassy for a temporary one. He had to put up with a lot of attitude at immigration coming back. He then had to apply for a new US passport once back in the US.
Besides, if the child is travelling on a UK passport, he/she would need to apply for ESTA I would think.