In addition to Buenchico's answer, Social Services can obtain the police record from the police if there is a Section 47 Investigation Children Act 89 being undertaken. Local authority's duty to investigate.
47.�(1) Where a local authority�
(a) are informed that a child who lives, or is found, in their area�
(i) is the subject of an emergency protection order; or
(ii) is in police protection; or
(b) have reasonable cause to suspect that a child who lives, or is found, in their area is suffering, or is likely to suffer, significant harm,
the authority shall make, or cause to be made, such enquiries as they consider necessary to enable them to decide whether they should take any action to safeguard or promote the child's welfare.
Checking on someone's criminal record is part of the enquires that are made.
This does not require the permission of the person being checked.