Just as an example, DS2 was in school for his entire reception year before he got a diagnosis. By day 3 of his reception year, the senco had identified that he had additional needs, within the week they had FT 1:1 support with him (funded by the school who then got additional funding through an outside agency), and the senco assisted us in applying for a statement. We got the statement about a month before we got a diagnosis from the paediatrician.
The school should be organising support NOW (should've already done so as soon as a need was noticed tbh!). They should be pushing for a statement NOW!
If he is getting outside agency help (OT, PT, SALT, etc), then he should be on SA+. He should have an IEP, which should be agreed upon by you as parents, and it should be reviewed regularly (at a meeting which again includes you as parent). If he is not getting this, insist in writing on a meeting with HT and senco to organise this ASAP. Take notes in the meeting and follow it up with a letter summarising what was discussed and agreed upon in the meeting.
Not having enough staff to support him is simply not an option. DS2's school had staffing issues occasionally, and we NEVER were told this. He required 1:1 support, they made sure it was there. That's what they are legally obliged to do - provide the support necessary for the child to access the curriculum.
Start a paper trail - get everything in writing. Any phone conversations or in person discussions, make sure you write up a follow up letter summarising and send it. It makes it very clear to them that you're making a paper trail and keeps them on their toes.
Sounds a bit militant, but you're the only one that is going to look out for what is best for your child.