It is confusing how the term Nursery is used in two different ways here, isn't it?.
The 15 hours a week funding for 3-4 year olds for Nursery is for what's called their Nursery Year. Different types of providers can provide these 15 hours - a Nursery Class atttached to a school, a private Nursery School, a Playgroup or Preschool, a Childminder, or a Day Nursery. The last two of these would be most akin to daycare in that they are often all day, the rest would be typically 3 hour sessions.
ALL of these follow the same curriculum (in England and Wales) called the Early Years Foundation Stage and are primarily focused on learning through play (as opposed to more formal learning through instruction as in school). There are certain Early Learning Goals that children should typically reach by the end of the school year in which they turn five and various steps along the way for providers to check that they are on track.
So basically the Nursery that is government funded is proper pre-school as you put it. The playgroup where I work (and where my daughter is currently doing her Nursery year) prepares children to be eager and interested and able to learn when they get to school. They learn how to interact with others and how to sit and listen to stories etc.
Now to get back to planning my small group activities for next week! If you have any further questions let me know.