autism assessment

As I explained above, it isn't true of self-financed private diagnoses either, providing the clinician or clinicians involved are qualified and registered to practice.

I financed my private diagnosis without a GP referral, the diagnosis was sent to my GP with a recommendation for treatment for co-occurring depression and a prescription was immediately sent to me.

Thats not true if you are referred for a private assessment by a NHS GP. I was referred to a private Physchiatrist  by my GP and was diagnosed with Aspergers. The Physchiatrist then referred my to Hampshire Phycologists for the full Autism assessment. The results were all copied to my NHS GP and used for my work adjustments

I'm on the NHS waiting list for an assessment, I should have at least a 3 year wait I think. 

Taken verbatim from the NICE guidelines for the assessment of ASD, "Diagnostic assessment, which in the UK uses ICD-10, is often within a multi-disciplinary team but at a minimum is by a qualified clinician, usually a clinical psychologist, psychiatrist or neurologist." From the document: Autism Recognition, referral, diagnosis and management of adults on the autism spectrum, National Clinical Guideline Number 142, p. 25.

Anywhere that has clinicians with relevant qualifications and specialisations for autism diagnosis, that are registered to practice with relevant medical bodies, should be perfectly able to give you a diagnosis that is valid. The thing that is necessary is the qualification of the clinician or clinicians undertaking the assessment, not whether the money to pay for it comes from the government (NHS) or from an individual (private). 

The vast majority of clinicians working privately also work for the NHS, discriminating between their private and NHS diagnoses would be both stupid and open to legal challenge.

To be more precise Psychiatry UK - Google them - does both private and GP referral, NHS funded, assessments for autism.