DS1 has Aspergers, he is 19 now and was not formally diagnosed until the age of 14. Sometimes it does take a long time. In our case, it was being passed around from one 'service' to another. He was given a statement at age 5 though - after assessment by an Ed Psych. IME, a statement was more key to accessing support in school than the actual diagnosis.
Do you know why DD doesn't have a statement? Is it purely because her SENCO doesn't believe she has SEN of any sort? Did she have a statement/extra support at her previous school?
You absolutely do have my sympathy OP - I remember the daily struggle to get DS1 into school & then DS2 off to his school (different schools as DS1 moved to a special primary school at 6 and then back to a mainstream secondary at 11).
However, I did learn pretty early on that;
a) lying to DS1, even a little white lie that seems harmless, would make matters worse. Even now as an adult, we still have to tell him the absolute 100% truth about every situation or he will worry & panic more than if we just said "don't worry, it's nothing" etc. etc.
b) don't ever say "we'll see" if there is a chance you might mean "no". DS2 & DD have always understood that "we'll see" means "maybe, maybe not". DS1 doesn't. It means "yes" to him. Always has, probably always will.
c) as my child, DS1's welfare was my responsibility. When he was in mainstream school, had he had a meltdown about going in to class and failed to calm down sufficiently, I would have been bringing him home with me. Yes it was a pain in the arse at times - in fact my career ended because of it as it left me completely unable to hold down a job for a few years. Yes, DS2 was often late for school. Fortunately, DS2's school were incredibly understanding and it was only during YR, as DS1 was at his special school by Y1 and had an arrangement to go in after DS2's drop off & be collected 10 minutes earlier.
Basically OP, having a DC with SN isn't always fun and mainstream schools will be able to do very little to support a child without funding for extra TA support. When DS1 went back into mainstream at 11, he did so with 1-to-1 TA support. It was his TA that spent 20 minutes coaxing him into lessons if needed - no class teacher has the time to do that.
I'm sorry, schools are so stretched. It does feel like you're always having to play the system to get anywhere.