You're being given the runaround I'm afraid.
The private consultant psychiatrist could and should have sufficient confidence in his own dx to be capable of presenting his diagnsotic findings without Ed Pysch or SALT input.
We also went private and our private cons psych did so. We also got a later NHS cons psych dx and that too was issued without Ed Pysch and SALT input.
As your private Cons psych also works in the NHS he's probably following a protocol that his NHS Trust has set down for NHS consultations - which can effectively prevent NHS cons from issuing NHS dx without Ed Pysch and SALT input.Different Trusts have different rules on his. But you are private so he's not bound by this.
Remember - you are paying him.
It goes against our nature but you need to be insistant and get him to issue formal written dx NOW!. Even it he has to write that the dx has been issued without the benefit of Ed Pysch / SALT.
If he continues to stall then contact your Local Councillor and complain that your LA will not permit your DS's diagnostic process to complete because they will not provide access to an Ed Pysch.
But it still shouldn't be necessary as my own DS's NHS cons psych also issued dx witho SALT input.
GP should have referred your DS to CAMHS - he should be on their waiting list.
But most importantly apply for a Statement yourself. Don't wait. You'll see an Ed Pysch etc as part of the assessment process.