Private Nursery (daycare) is from Birth to five and is mostly used by working parents
Pre-school is from 2.5 years to 5, is generally community based sessional care (generally children go for 2.5-3 hour sessions, run by a voluntary committee.
Nursery school is from aged 3 - almost 5, may be attached to a primary school and may provide either full time school hours or just morning/afternoons
All of the above usually accept the nursery education voucehers but you may have to pay top up fees at the first 2 (and the third if it is at a private school)
Some children will stay at a daycare nursery until they start school, some will go to pre-school then nursery school, others will stay at pre-schol, it is up to the parent and how things suit them.
Primary School - Generally starts the academic year in which a child turns 5 but you can defer going to school until the term after a child is 5 if you really want to (most don't)
Junior School - Some areas have separate junior schools from year 3 onwards (the year a child turns 8)
Secondary School - ages 11 - 16/18
FE/6th Form College - ages 16-19 and beyond