I had a very very similar situation with my late summer born DS and I can tell you what I did, warts and all.
So he presented very similarly to your child, mostly 18mths level in his EYFS assessments. Had very little language. Was on wait list for autism assessment (finally received that diagnosis two months before he started school). Our LA didn't refuse to assess, but did refuse to name specialist, which I knew would be a huge waste of everyone's time. He would not cope with the noise in the room, wasn't toilet trained and clearly needed 1-1 input to be able to access any learning at all.
I made the decision to keep him in nursery for an extra year while I fought out his EHCP with the LA, honestly IMO the wrong placement is way more damaging than no placement. And I knew he needed specialist, the visit to the specialist only further confirmed what I already knew. So we stalled, even though his care was crippling me and I had a newborn at home. I read the IPSEA website like a demon, bought a box file and started building my case, because I knew it was going to tribunal. And it did.
I am more proud of that tribunal bundle I submitted than any essay or thesis I've ever submitted in my life, it was meticulously researched, referred frequently to the Equality Act and had numbered supporting evidence. It was a beast of a document. It worked.
The July before his compulsory school age year was to begin, the LA conceded and named our preferred placement.
He's ten now and still at his specialist provision and probably will be until he's an adult. He loves it there, and he's treated with respect and love. That year of hell was absolutely worth it.