You ideally need to find somebody who can give you pretty much what NICE guidelines recommend, which is a multidisciplinary assessment.
I do this, but I'm not in your area (though I trained at Reading University, so I'm waving at the lovely Berkshire!) so I feel I can mention that without being accused of touting for business!
You ideally need a doctor, psychologist and speech and language therapist (if you read the NICe guidelines, you'll see) who can conduct an assessment using interview and observation. So for example, we use the DISCO assessment for interview and the ADOS for observation. We also do a sensory assessment and advise on that, which you might find useful - so try to incorporate that or perhaps see an independent OT for it.
There are a couple of people who assess for autism acting alone on a national basis but as far as I know, they tend to use one tool or the other but not both (to conduct an ADOS, you need a second person qualified to administer the test who can act as 'second rater' to make sure the scores are fair). As far as I can see, this just doesn't fit NICE guidelines - one person cannot be 'multidisciplinary.'
Their opinion may or may not be accepted - but when you are paying a lot of money (and I know what they charge - it's over and above what I'd expect to pay) then you need to find something as close as possible to NICE guidelines. I seem to think there's a place in Oxford, which is obviously bordering Berks, but then again I think it's just psychologists so not multi-disc.
In our case, we don't have a paediatrician on board - but people often come to us because of the waiting list. Our reports are sent to the paed at the hospital and we've never had a problem getting them accepted (so the paed's opinion completes the triad of professionals required by NICE). You might not find the exact combination of SLT, psychology and doctor in the private sector (or you might!) but provided your doc is prepared to accept a high-quality report, you ought not to have problem.