@Alltheholidays "l'm always astounded with posts like this! If I had a child who was hurting other children and it was obvious the teacher didn’t want him in the classroom because of his behaviour, i I would be thoroughly ashamed, and not looking to blame everyone else."
Bully for you. Sorry you would feel thoroughly ashamed.
Fortunately it sounds like OP has a wider emotional spectrum than you, and well she might need it as it sounds like she will have a heck of a fight on her hands to get her son the SUPPORT he needs; the ASSESSMENT his nursery denied him; the RESOURCES to help meet his (as yet unquantified) special educational needs; the TRACKING appropriate to identify specific scenarios and triggers leading up to inappropriate classroom behaviours; and the EXPLANATIONS she and the school staff need to understand what can be done to help his behaviours improve.
Classrooms can be overwhelming places for July born reception children. Particularly if the teacher is a give-no-quarter bully who makes no effort to adjust their teaching methods.
No, biting and hitting is not an appropriate behaviour at school or elsewhere, but it's far from unheard of in a cohort of 4 year olds, even developmentally neurotypical ones!
There is so much a Year R teacher could and should be doing with the parents to help a 4 year old settle in - and not be a risk to his classmates. Threatening exclusion without a conversation between them, the parents and the school SENCO isn't one of them.
It sounds like this teacher is utterly out of their depth in a mainstream Year R classroom of 30, particularly after COVID.
OP you mentioned that the teacher had recently moved from the private sector. It sounds a lot like their teaching history has been exclusively in schools with small class sizes and pre-screening out any special needs. If the teacher is committed to teaching in the state system, they need to get up to speed PDQ with how to assess and support the full range of behavioural issues of children who can quite reasonably be expected to remain in the mainstream system with appropriate support.
It is quite unusual for teachers to leave the private sector and move back to state. Unless this teacher is fully committed to the broad spectrum of state teaching requirements (which it hardly sounds like they are!), I would be suspicious that the quality of their teaching didn't even meet the required standard to retain their original job in the private school.
There are a couple of big red flags around this teacher's handling of the incidents to date.
OP, can I suggest that you ask for a meeting with the school SENCO & head of Year R/head teacher to review the incidents to date, scenarios leading up to them, and how they were handled in class. The teacher should have documented each of them if they resulted in any agression to another child. This should give you some initial answers - either on events leading up to poor behaviour and how these could be addressed differently for your child in future (the beginnings of an EHCP plan, or at least referral document), or otherwise on the inadequacy of the teacher's ability to manage & document a standard mixed Year R group in a state school setting.
If it's the first the head has head of it, you will have a significant answer. You can then begin to backtrack & work fully with the school to put appropriate steps in place to manage DS' behaviour to ensure it doesn't get to the stage these incidents result in agression in future.
School budgets are extremely stretched & extra TA support or 1-2-1 might not be possible as easily as SENCO guidance suggests it should. But by being open & co-operating with the full school staff, at least you can determine whether it is one duff teacher's opinion that the easiest way to make their life in the classroom simple is to kick your son out. Or whether this is a more pervasive attitude from across the school. In which case get meeting minutes and take it to SENDIAS and your LA to ask for support getting an assessment and finding a local mainstream school who might actually have the skills they should to support a 4 year old who is hardly developmentally abnormal, but may need more support to adjust to school than he's currently getting.
Good luck 
And do ignore the virtue shamers who are fortunate enough not to have had to support a child with some additional needs in adjusting to school! You sound a very committed & sensible parent and I'm sure working strongly with the school will enable you all to help your son meet the high standards of polite behaviour you expect at home in this new, demanding, overwhelming classroom environment as well.