Cost estimates sought in this question are not available. However, I can report that:
(a) It is not possible to separate out the cost of conducting an educational assessment on a child or young person in juvenile custody from other delivery as the cost is incorporated into the hourly rate paid to staff working in juvenile custodial establishments. As custodial establishments vary in size, the larger ones will carry out proportionately more assessments and will therefore benefit from economies of scale in price negotiations within awarding bodies.
(b) Local authorities are currently not responsible for assessing the educational needs of young people in young offender institutions because local authorities do not currently have responsibility for providing education and training for young people in juvenile custody. The current Apprenticeships, Skills, Children and Learning Bill proposes legislation to change this to make local authorities responsible for securing education and training for persons in juvenile custody.