In England the majority of DC with SEN are supported at a school level in mainstream schools. With the school making adjustments and providing interventions and SEN support.
For those that need more than this you can apply for an Education, Health and Care Plan (EHCP), which is a legal document that, among other things, sets out a pupil’s needs and the provision they require to meet those needs as well as the setting they will attend.
Some DC with EHCPs are in mainstream schools, some in special schools, some have education otherwise than at school (EOTAS), and some attend specialist resourced provisions (what they are called varies areas to area but that is what they are called in Hertfordshire). The last sounds the closest to what you describe, but each provision is set up slightly differently. The majority expect pupils to attend mainstream classes some of the time.
As soon as you/DS are resident in the LA you will be able to apply for an EHCNA (a needs assessment you first request before the LA decide whether to issue an EHCP or not).
Not all mainstream independent schools are supportive of SEN, and if they are they often charge extra and the cost of SEN support can mount up quickly. Not all specialist schools accept self funding pupils and where they do the fees can be large. So, if you go down that route have a frank conversation beforehand. If you get an EHCP independent schools can be funded via that.
You can search for schools here.
Reading the SENCOP will help you understanding the system better. IPSEA and SOSSEN are charities that support parents with SEN matters, looking at their websites will help too.