Daphne Keen is very highly regarded in her field of expertise, on a par with Lorna Wing Centre and Judith Gould.I don't think any school or LA could challenge their diagnostic credibilities, simply because they also accept NHS referrals. The NHS are not going to refer to a professional then refuse to acknowledge the validity of their dx.
I've only encountered the 'you paid for that dx so it's not as credible as an NHS one' viewpoint on things I've read on MN over quite a few years.
Much of it comes from playground gossip 'Her kid only got dxd because they paid for it and it's an excuse he hasn't got that, he's just a little expletive, she's a crap parent. Our kid has a proper NHS dx.'
Obviously, totally ignore anything you hear from those uninformed sources.
The ones to be concerned about are the parents who report that either School or the GP or the LA will not accept private dxs.
GPs cannot 'rubber stamp' any private dx, because it has not been carried out by an NHS professional, unless the professional concerned also undertakes work for the NHS , like Daphne Keen, Lorna Wing Centre etc.
Schools - Usually SENCOs - This could be because the SENCO knows the diagnostic team do not also carry out NHS dx and is refusing on that score or because parents have paid for a comprehensive screening test which outlines all the pointers for a dx so believe they have an actual dx, when in fact all they have is a 'strong likelihood indicator' of their child having that condition. even the Lorna Wing centre do this sort of test and I've seen someone post here very distressed that her school wouldn't accept LWC's dx, only to discover it wasn't a dx, only the screening test, therefore school could not action any interventions as it was not a full dx.
LAs This comes up a lot when people apply for EHCPs. 'The LA will not accept the private dx.'
I never went through that process, so again from what I've read here, it seems as though it's a delaying tactic by the LA as they can be very oppositional do deal with. The poster Goldmandra I think did outline a way to prove a non-NHS dx HAS to be accepted by the LA, but I can't remember what that was, so if anyone is in that situation, maybe drop her a pm.
TL:DR There is no way a GP, School or LA could refuse a dx from a professional who also dxs for the NHS because a parent paid to have their child assessed by them privately.
They would in effect be challenging that professional's or the diagnostic team's credibility by saying NHS and private clients were assessed differently by the same people.
Imagine the storm caused by suggesting the same diagnostic team assessed NHS clients fairly and accurately, yet gave a dx to anyone who paid them privately.
I don't think you'll have any problem going to Daphne Keen, but by all means ring her Secretary, I'm sure they must know of the situation and be able to put your mind at rest.
Good Luck for the assessment 