Have the nursery instigated any of the Early Help meetings?
You also can request an EHCP for your child even if they're only in nursery and you can do this yourself on your local councils gov page.
This will trigger a needs assessment, although you can also pay privately for an educational psychologist to do an assessment if wait lists are long. There are legal time frames local authorities must abide by though.
It sounds like a specialist provision might be beneficial to your child and many will only take them with an EHCP so I would get on that ASAP but the type of specialist will be dependent on whatever primary need is assessed during a need assessment. For example my son was like your son but was significantly speech delayed so his primary need was social communication and interaction. This means that specialist schools would be in that area and not social emotional and mental health, moderate learning disability or physical health needs school unless it is a school that covers multiple primary needs as some interlink.
I would advise instigating the EHCP process as soon as possible as you may have to go through the lengthy appeals process as your local authority are not your friend when it comes to your child's education, but there's absolutely no reason to believe despite your child's current struggles that he can't go to a school or receive an education that is suitable for him.
The school we are trying to get my son into focuses on life skills and play based learning with the options for academic qualifications if it meets the child's SEND needs and aptitude, which so many go on to do with the right support.
It's a scary road, but once you take the leap you'll find strength to fight for what your son deserves and it becomes a lot less daunting when you realise there are so many support services out there to help you stay on track.