We've never seen a dyslexia assessor, but DD's dyslexia and hypermobility were both picked up by an Ed Psych.
They spent best part of a day with DD in Y3, did a huge range of tests and a very comprehensive report. It meant that we didn't just know she was dyslexic (that was blatantly obvious to everyone by the time she was able to be tested), but knew exactly where her strengths and weaknesses are. Had a second set of tests and new report in Y6 that was almost identical and confirmed these weren't things that were going to resolve with age or teaching.
There was also a list of things that needed to be put in place at school - sitting at front of class, moving to a laptop for as much as possible - and a list of suggestions of things we could try... coloured overlays (didn't work), theraputty (helps a bit) etc.
We did briefly discuss potential ADHD as there is an incredibly strong family history, but you need a psychiatrist for that, and it wasn't what was being looked for at the time. Eventually that was causing more problems than the dyslexia and we got that Dx in Y9.
Other things an Ed Psych could potentially look for - dysgraphia, dyscalculia. Might be able to advise on processing issues. Every dyslexic will have their own specific profile - DD has VR/NVR on 98th centile, and working memory on 7th - that kind of spikiness if probably at the root of the problems. Her processing speed was 50th centile which is average, and the disconnect between that and the other scores means that it causes her problems because they're not in synch. A child with all scores in the same ballpark will find school a lot easier to navigate.