It varies by area. Some gps will just refer to camhs as a diagnostic authority, so it depends where you are.
What is the paed doing? Is he reviewing in 6 mos? (fairly usual if not enough evidence)?
Do you have a diary/ list of behaviours? Sometimes a bit more time crystallises school/ nursery's thoughts. Have you sat down and discussed with the senco? Have they had area inco in to give advice?
Private assessment will at least tell you if it is worth pushing the NHS to take you seriously...
At the mo I would think about what a dx means? Does he need support at school? Are you looking to go on early bird parenting course or similar?
What difference will a dx make for you in the short term? Can you access this stuff without a dx?
It does often take a long time for dx to happen, especially where you have professionals (of whatever type - ie school) saying there is no problem, so it's a matter of getting your ducks in a row to prove need, or to pay and get it done privately if you need the certainty. The only reason that they wouldn't accept the private dx is if it was going to cost them money - so if school are saying he's fine and you aren't thinking of applying for SA, then there is no real need to worry about acceptance of dx.
A lot of folk read up and start using methods suitable for the dx in any case - in that hinterland is he/ isn't he period. If he is, they might help, if not, no harm done.
I think I would be concentrating on getting my ducks in a row and turning up at the next appointment with fairly irrefutable stuff that means further assessment is necessary - for whatever reason, what you turned up with this time wasn't deemed enough. Or I'd pay. We've done that in the past, too.
And I'd work out what you are hoping to follow an actual dx. Sometimes it's pretty much nothing... That's not to say it isn't worthwhile, but often there is no actual concrete difference, just a label. It does become easier to fight for what you need with a label, but it's helpful to know what you need, and you can start fighting even before, iykwim...