Rabbit the advice we had about assessments came from the charity that we organise the assessments through, we pay but those who cannot afford it do not, they work with plenty of children from state schools. And they are 60 miles away in the sticks because London Ed Psychs are exorbitant and completely booked up. My DDs last assessment had been pre 16 and we were advised we would need a further assessment, not a full Ed Psych report but a supplementary report produced by a teacher assessor, their teacher assessor. Presumably the advice is available online and is something you will want to explore with the Learning Support set ups at the unis you visit for open days.
I am struggling to recognise the slightly dizzy single SEN teacher at my DDs private school as some sort of savvy scheming SEN fixer. She is lovely enough for giving reassurance and occasionally even the right advice on how to tackle essays etc but she wasn't even sure if it was me or her who originated the form the teacher assessor signed for the exam board, and in the end I had to fix it.
I am not extrapolating anything, my whole point is that state school provision varies enormously. Our LEA just happens, probably as a result of parental crusades, to have processes in place that ensure good provision for pupils with special needs, but I know that does not extend beyond our borough boundaries, and nor does it extend to ensuring a good standard of education for all it's pupils, or even enough places for all it's pupils but that is another story......
One thing I have learnt from my and others experiences is that in state or private schools if you have a child with SpLDs you have to be constantly on the ball, acting as facilitator, investigating options, correcting misapprehensions etc etc etc. it shouldn't be the case because pupils without the advantage of a proactive parent definitely lose out. The smoothness with which her firmed uni has started the process of putting support in place and the Student Loans process has led to her being granted DSA and her needs assessed has been a complete culture shock for me.