Historically, an establishment calling itself a nursery would probably be privately run, and would cater to children from just over bith to almost school age, whereas typically a preschool or playgroup was really meant as a preparation for going to school, and thus for older children (over 2 if not actually over 3). Preschool education used to be provided in "playgroups" but in a move to try to encourage these schemes to take themselves and their educational role more seriously, the name preschool is now preferred over "playgroup" by regulatory bodies (like Ofsted and the local education authority).
That said, the names are becoming increasingly unspecific. Many schools run an associated "nursery" which is targetted exclusively at children over 3 intended to attend that same school. "Playgroup" is increasingly used to describe parent and toddler groups (which are mostly intended for under 3 year old children).
If you are trying to decide on a preschool or childcare place, the best thing to do is to ignore the nomenclature, and simply treat each provider as an individual service provider, asking them what they think they do and what they can do for you and your child.