Also don't get mixed up between Private schools and Private Nurseries.
Private schooling (despite the impression you would get from MN) is really quite unusual in the UK. My understanding is between 7 and 9% of children are educated privately. So, overwhelmingly, children in the UK have state funded schooling.
However, prior to school age, the majority of childcare is delivered by what is know as PVIs (Private, voluntary, or Independent providers). These can be privately run businesses, Local Authority run Children's Centres, or provision in Church Halls or Parish Centres or Scout huts.
The Government talk about 'free places' but what they mean by that is they will (partially) fund hours for parents to use at any of these providers (or Childminders)...... note, it will still cost you, but that is a whole other thread.
Every child is entitled to 15 hours of Early Education Funding (term time only) from the term after they turn 3. There are a small number who are entitled to 2 yr old funding, and then working parents can claim up to 30 hours (term time only) in some circumstances.
There are some 'stand alone Nursery schools' which tend to operate school hours and have a teacher there. There are some schools that have classes for Nursery children as part of their school.
Names are interchangeable - Nursery / Daycare / Pre-school / Kindergarton / etc. Don't be fooled by that.
In summary - it is really complicated, yes 
Might be easier if you say what you are looking for thatn for us to explain everything.