When I worked in primary which admittedly was as quite a long time ago
All children would have a yearly parents evening and a report. For reception the report and the parents evening would cover the early learning goals which include social and emotional learning.
If a child is struggling at school - either socially or in terms of progress - at reception I would expect it to be discussed with the parents then.
That's level 1.
At level 2 the teacher identifies that the child is having some difficulties either socially and emotionally or in terms of progress. They might then contact the parents and tell them that their child is having additional input. So maybe a small group doing something like social skills run by an ELSA www.twinkl.co.uk/teaching-wiki/elsa
Often at level 2 the parents will want to talk to the school. At this point some parents can be incredibly resistant to getting their child help. Strictly speaking the parents do not need to give permission for the child to attend this group as the school has decided it is educationally appropriate but some parents absolutely kick off at the idea that their child might need additional support.
Level 3 is then pulled into play, in theory if level 2 hasn't fixed things but in practice some children are so obviously in need of support they go straight to level 3. This is where they get individual intervention and assessment from external specialists. So the school may ask for support with (for example) a deaf child from the LA teaching service for D/deaf children.
Depending on how a child with autism presents, many do go straight to level 3. The school will then set up a meeting with parents and suggest further support/assessment.
Many parents will react very defensively. It's a real skill running those meetings as often the parents are very upset that their child is being deemed "substandard" and often you get a lot of complaints afterwards unless handled very delicately.
You will also get the parents who know their child doesn't fit the norm but "don't want them labelled".
It's also tricky because teachers cannot diagnose so all you can really say is I think they need assessment/more support.