I've been applying for preschools recently and my understanding (based on what is around here) is that there are few different types.
Some are attached to a primary school and are run by the primary school, these often only accept DC from age 3 onwards, rather than from 2.5yrs.
Some are in church halls or civic halls, ran as non-profit by a charitable organisation/church and usually accept from 2.5yrs onwards.
If a preschool accepts from age 3 only, they are often non fee paying because all the DC attending are over 3 they therefore will be eligible for government funding for their 15 hours at preschool, so the parents don't have to pay.
If the preschool accepts 2.5yr olds, then those 2yr old's parents will have to pay fees (as they are too young for the free 15 hours care) and so that preschool is not a non-fee-paying preschool.
I think all preschools offer sessions, so mornings 9am-12noon and afternoons 12noon-3pm and you sign up for as many sessions as you want. If they are church hall type preschool they don't always offer sessions on everyday, so maybe only on Mon, Wed, Fri for example.
When you say ' a normal nursery where you can choose when to send your child, i.e. 3 mornings, 3 afternoons, 4 full days etc?' I think you mean a childcare nursery, a business, run for profit that provides full day care for babies through to school age children. They usually have a 'baby room', a 'toddler room' and a 'preschool' room and children move their way up through the rooms until they leave private childcare and start at school. Once your child reaches 3, some will let you use your free government 15 hours against the cost of their childcare, but many (particularly in London) won't do this these days and so you end up paying for all the hours your child attends.
I hope some of that makes sense (and I didn't get any totally wrong!).