There is a HUGE difference between the difference between Private and state schools but the Nursery situation is vastly different.
Only something like 8% of children go to Private schools.
It is a HUGE financial commitment, over 14 years of schooling.
However, the overwhelming majority of Nurseries are 'Private' or Voluntary/Charity / Community. As in, they are private businesses, not run by the state. Though, they all take state funding to provide places that the Government like to pretend are free to parents.
There are some 'Stand Alone Nursery schools' that are maintained (state run) and some schools do have Nursery classes.
These have historically been for dc in the year before they start school (although, just to confuse, many of these take younger dc now as well).
PVIs (Private, Voluntary and Independent) are more likely to operate all year round (though not all do), and are more likely to operate from early in the morning until early evening (again, not all do) whereas school Nurseries tend to be school hours only and term time only.
In theory, ALL Nurseries follow the same curriculum and ALL Nurseries should differentiate and should offer the same support for children with additional needs.
In practice, some are far better than others, but it is individual Nurseries rather than one 'type' or another.
In my LA, nearly all Private schools have a really poor reputation for supporting dc with additional needs once they get to school age. I perceive there is a reticence to support children with additional needs in many private schools, but wouldn't like to tar a whole industry with the same brush.