There is no definitive answer (well, except the Primary School admission date) because there is so much cross over with language.
Things are also different nursery to nursery - it isn't even geographical.
Don't get hung up with the names, as the same name can mean different things.
So, as others have said, there are Nurseries that are 'day care' - originally for working parents. These are known as PVIs (Private, Voluntary or Independent) and you pay for a place, or, when the time comes, use a 'funded place' for 15 hours.
This was originally 'Early Education Entitlement' to encourage uptake of Nursery places to resolve the inequalities of children starting Reception, but Government have messed about with the funding for political gain, and there are all sorts of different fundings now available. In most Nurseries, you will have to pay a top up (called other things) to make up the difference between what the Gvmnt funding is, and the actual cost of the place.
So, as the places are not quite free, lots of people who don't need day care (eg SAHPs) still get their dc a place at the PVIs from the term after their 3rd birthday. So, if your child is born Sept - Dec, they get 5 terms of Nursery before school. If they are born April - Aug, they get 3 terms.
PVIs also cover extended playgroups, in Church Halls, Scout Huts, Parish rooms, etc.
Every single PVI runs independently of the others. Most are businesses, some are community run by a committee (see Church Playgroups). Fee paying / Independent / Private schools often have a nursery attached, too. You can go there even if no intention of going to the school.
ALSO
There are 'Stand alone Nursery Schools' which are not attached to any school, but have a HT, and teacher(s) and Nursery Nurses / Early Years Practitioners) who are employed by the Local Authority.
You don't need to pay here.
Some do 2.5 days, some do all mornings then have a different cohort of children in, in the afternoons.
However, this will be in school hours.
Because there is a teacher in charge, the ratios are much higher. They only need 1 adult for 13 children.
In a PVI they need 1 adult for 8 children.
ALSO
Some schools have a Nursery class, on site.
Same re cost, hours, and ratios as SA Nurseries
But, ANY of these can use names like 'Nursery', 'Pre-School', 'Kindergarten', Nursery School if they want to. Do NOT think that will describe the provision