My eldest was like this. Started school at 3 and they quickly talked about permanent exclusion. At nearly 19 they're running an international team of technical bods in a large corporation, earning a very decent salary and just moved into their own accomodation. Also have a list of diagnosis as long as my arm but they're my wonderful very individual DC.
My thoughts to add to the many constructive bits of advice... Keep being your childs advocate, you already have been by pushing for assessment at nursery.
Its time to up your game - push back they can't just dismiss your DC. Start a decent paper trail, did back through any old emails and texts you have from nursery to give history to your concerns. Document every conversation with the school. A simple summary email of conversations and any actions suggested in them creates a valuable dated document that can be refered to in the future.
Document every conversation, every behavioural issue, every intervention/ behavioural management technique you try and they try.
Ask the school if they would support an Education Health Care Plan application in parallel to the steps they're taking. IF you can get your DS's behaviours assessed in a setting its easier than post exclusion. If they don't agree, apply yourself. If you get an application accepted your son should be able to get an Educational psychologists assessment in school and they will be able to advise if they see special educational needs or if there is a developmental delay etc.
If you can avoid pulling your child from school, force them to do a formal push in writing. It sounds like a technicality but then the council have to actively be involved in sorting out alternative and more appropriate education. They may also help with things like transport logistics to another setting. Shen you have multiple children at different settings with similar start/ finish times is essential for your sanity and purse strings. I had three young DC at three settings at one point - complete logistical nightmare.
If you pull then the council can be a little slow in dragging their heels to sort the right new placement. Don't be afraid if the next placement isn't right either. We did 5 primaries (and moved counties twice) all in the search for the right fit. When we found it life really started to fall into place.
This isn't necessarily a disaster. Its distressing and the strong Mum instincts kick in wanting to defend your child - but the fact the school are not the right fit and they've picked this up early could help you find that right fit and have a more contented child much faster. Whats round the corner could be a much better solution.
Good luck - the special needs boards can be very helpful. Masses of knowledge on what goes on with getting assessments and shoulders to help share the burden of stress going through it.