NHS or Private diagnosis for claiming PIP?

We have 20 year old twins who we suspect have Aspergers.  We have been to a private psychologist (found from the Directory of Chartered Psychologists) who has undertaken a part assessment and diagnosed them as having  Aspergers. The twins firmly dispute her diagnosis but at present they sleep all day, are up all night, rarely leave their bedroom and are not succeeding in any attempts to progress with their lives.

My question is would a full diagnosis from a private psychologist be eligible supporting information for claiming PIP or would we still have to undergo a diagnosis with the NHS before completing the mandatory PIP assessment.  Obviously to go private is a lot of money and we don't want to spend that if we have to end up going down the NHS route. 

Has anybody had any experience of this or have any advice please?

Our local authority is Hampshire County Council if this has any bearing on the answer.