My dyslexic /dyspraxic Dd got extra time in exams under the new regs, the key criteria was being in the below average range for processing / working memory. I forget which percentile she was, 9th?, but I seem to remember that the below average range is below the 14th centile ie 86% would score better. However that is the criteria which automatically grants extra time, the SENCO types in the scores and assuming an Ed Psych report and evidence of need is also available then extra time is given. Obviously evidence of need and Ed Psych reports can be reviewed on a case by case basis but certainly it has become harder to succeed with that sort of special consideration. The new regs actually discriminate against very bright dyslexics, as OFQUAL acknowledges, since their processing /working memory scores could fall in the average range and still represent a disability, the latest regs were to placate the Daily Mail readers perception that getting a diagnosis is easy, and are a move away from the principle of levelling the playing field. Ed Psych are furious because it actually grants extra time to pupils whose processing scores are below average because that is what you would predict for someone of below average ability. It is certainly NOT the experience of my DDs peers whether in state or private schools that getting a diagnosis is easy.
My DD is 18 so I have experience of children with SpLDs going through the school system. As others have said some private schools are very bad at dealing with SpLDs, some are adequate and some specialise in it. I simply do not believe that a selective private school that does not specialise would have 50% of pupils getting extra time. The private sector around here (London suburbs) educates around 30 % of pupils and all identify SpLDs in around 10% of pupils, which reflects the proportion in the general population regardless of ability. Selection tends to be on ability and potential rather than attainment so you would expect them to find that proportion in the pupils they select. Of course there are specialist private schools, our LEA actually buses 30 boys to one and pays their fees because it's own units do not serve their needs, where 100% will be diagnosed.
In the state sector the problem tends to be that the resources are focused on those pupils whose attainment is below average, a pupil whose attainment is above average, even if it is below their potential will struggle to get the benefit of limited resources and as others have said often those parents, if they recognise the problem, end up paying, whether for an Ed Psych report, or for private education. To that extent, yes, money talks.
There is limited coverage of SpLDs in teacher training so you are also dependent on schools recognising the need for it to be covered in further development, and so the ability of teachers to recognise SpLDs and for schools to have processes in place that will lead to diagnosis is very variable. That is true for both state and private.