There have been many Laws passed for the employment of children and one main point is the minimum age. If an employer was found to have underage workers they would face stiff penalties.
A full birth cert. was not necessary as many families would find the cost prohibitive, so Register Offices could issue what was known as a Short Certificate which contained very basic details, but were sufficient to fulfill the employment requirements, at a fraction of the cost. I believe 3d and 6d were common charges for different types.
Regards,
Colin