Referral or Private ASD testing

 Our lovely 6yr son has been suspected of sitting on the ASD spectrum since he was 3.  We had a GP referral last June that got rejected before he was seen by any specialists as he  was deemed not serious enough.  His school who have been  supportive were chasing a 2nd referral through the school nurse system however this is taking so long they have asked us to go back to our GP .  We have the all about me forms filled in by them and us, what else can we do/ say to try and get him seen for a true assessment.  I am concerned  he will be rejected again  and I am beginning to feel that we have to consider a private diagnosis.  Has anyone resorted to this, in particular in Nottingham.  Is a private diagnosis likely to be accepted by schools and the local authority.

Many thanks in advance