Yep, J had no dx, was told by EP that he'd never get a Statement, applied myself (he was 3) and got turned down, filled in appeal form, they backed down, gave him 24 hours which have now been upped to 33. The ADHD and autism dxs helped up the hours to 33 but he already got a substantial amount before dxs.
I never had to pay for any private reports, and the ONLY ONLY ONLY criteria for a Statement is that the school cannot meet the child's needs from their own resources. Schools are resourced in different ways in different areas of course but that is the basic criteria and only legal criteria, though LEAs try to make up their own illegal ones.
e.g. if a school is expected to fund 20 hours of 1:1 from their own resources, a child will probably only be eligible for statementing if they need 22 hours. I'm simplifying a bit. The important thing is to make sure that the school is doing anything possible and everything that they're supposed to be doing otherwise a Statement is very unlikely as the LEA (or appeal panel) will simply say that the school should be doing more from their own resources.
Worth asking the LEA how many hours the school are expected to provide from their own resources, then you know what the school SHOULD be doing.