From FlyBE's requirements (much less strict for Domestic travel):
Children and domestic travel
Children under the age of 16 do not need to show ID on domestic
routes. The adult they are travelling with can vouch for the
infant's/child's identification, providing the relevant photo ID is
shown (by the adult). On International routes a passport will be
If the child is traveling unaccompanied and aged between 12-16 years,
they will be required to produce photographic ID.
So it seems like they're register that there was a child travelling with you, but would rely on you for the identification. No other document is required.
Again, this is for domestic only.