Forgive my ignorance, but why is the assessment taking so long?
I imagine it must be due to the lack of resources and waiting lists and all the administrative procedures that have to be followed - i.e. a school needs to carry out an initial assessment, decide whether there is a basis for a referral, write to a relevant department requesting a referral etc. It all takes time to process. The assessment needs to be carried out by a certified educational psychologist, a full assessment takes about 3-3.5 hours and there are not that many psychologists that education authorities can afford. It may be even more complicated if a school is an academy - I am not suggesting the OP's school is, but I'm just wondering how the process works with academies.
Incidentally, when done privately, it's usually quite quick. It took us 4 weeks, but only because the particular psychologist we wanted to see was away. A referral was not required. I know someone who had the whole thing sorted in a week.
Whether additional time in the exams is granted or not, has nothing to do with whether the assessment is done privately or through the school/LEA. The only thing that matters is whether in the Educational Psychologist's view a particular child's degree of dyslexia is severe enough to warrant allowing them the additional time. A relevant recommendation is a part of the report and it is the report (a copy of it) that needs to be submitted to the secondary school in question as a proof that a child needs to be granted the additional time in the exam.
I imagine when the teacher said that going private did not necessarily help getting extra time it meant exactly that - it is not about private vs. education authority, but about whether there is a justified need for the extra time. Going private just lets you find out much quicker where you stand with this.
NB. Initially, our school wouldn't even consider referring our DC for dyslexia assessment - he was top of the class for maths, quite good at reading but could not spell. It was only on our insistence that they agreed to a screening test done in-house and it came back with a 'moderate dyslexia'. We were told his performance at school was too good to qualify for a referral to LEA for a further assessment, but at least we knew what we were dealing with. We had no choice but to go private and very fortunate we were able to do it.