We have no idea what is going on behind the scenes at school. As someone who works in a school myself, I can almost guarantee they will be monitoring the children's wellbeing closely, due to their chaotic lifestyle. However, we as the general public have no right to such information. The school are hardly going to announce it to the media!
As for Archie, whilst he definitely displays some potential learning disabilities (speech delay, potential early indicators of autism), it will all be considered as age appropriate right now. Sadly so many children start school these days with incredibly poor speech, difficulties regulating their emotions etc. School will be looking at progress. If he fails to progress in the way you would hope or expect for a just turned 5 year old who has just started school, that's when outside agencies will be brought in.
If his delays are purely environmental, rather than because of a diagnosable condition, nursery/school will do him the world of good and progress should start to be made. There's no denying he still struggles with speech, but I have noticed some improvements in recent blogs, the longer he's been at nursery. It's what he desperately needed.
I'm sure he is being monitored by nursery, and assessments will be carried out, but it is incredibly difficult to get an EHC plan (new money 'statement') these days. I have known far more extreme cases than Archie to be refused an EHC plan, due to cuts. Unfortunately, I really can't see him getting such support. His nursery should be referring for speech therapy (maybe they already have, I can't imagine the Radfords making such information public) but a diagnosis of autism often takes years. It's only usually the most extreme cases (non-verbal, violent) who are diagnosed before they even start school. Unfortunately, Covid will have pushed things back even further
Right now though, it's highly likely Archie presents the way he does due to his home environment, rather than medical and cognitive factors. Maybe he presents differently at nursery, where his individual needs are catered for and met?