The main difference between school nurseries and day nurseries is the level of qualification of the staff. A school nursery must have a teacher in charge and other staff are level 3 (often NNEB trained) while in day nurseries the person in charge is often a level 3 supported by level 2 and sometimes unqualified staff. The downside of this is in a school nursery there is a ratio of 13 children to one adult where in a day nursery the ratio is 1 adult to 8 children (3 or over).
Where I live - in Paris - yes, it's wildly different.
Pre-school is taught by qualified primary school teachers (ie 5+ years of university) according to the NC. Quite a few of the teachers at my daughter's pre-school are doubly qualified as primary school teacher eg in the US and in France.
Nurseries are staffed by people with a professional qualification in childcare which they take aged 18 in lieu of the baccalaureate.
Ours is very different - it has qualified teachers and it's more focused on the transition to reception than a day nursery would be. Our school nursery is right next door to the reception classrooms and the nursery children go into those classrooms for story time, they go to assembly in the main school hall with the rest of the infants, they take part in sports day with the reception classes, they share a playground with reception sometimes. Towards the end of the academic year they play in the main playground with infants to get them used to it for September. They also take part in all the infant productions.
Ds started today....I can see how the difference in ratios might have an impact. Was expecting there to be more structure at the school nursery but there seems to be less. Not sure it matters as happy for ds to play but was surprised.
Chipmunks I think it's changed quite dramatically in the last few years. my 8 and 7 yos went to preschool/school nursery and learnt things like jolly phonics and eventually how to read and write and add up, which their private nursery also did at the time for the over 3s.
But my 4yo, in a preschool/school nursery, only plays, and does a bit of counting. No phonics at all. Nothing formal. The odd picture with a letter attached is as formal as they get - it's totally different from my dd2 who was taught to read at the same stage. The preschool curriculum recommendations have changed a lot just in the last couple of years. Now it's all about play not formal learning.