Hi there, I live in Herts. They go to the school nursery at the points that the council told you, but you can get funding for them from the term after their third birthday which is the same nationwide.
So... most go to pre-school first and have this funded, then go onto school nursery at the appropriate time.
Hope this answers your question, I know it's confusing!!! As an example- my son did pre-school from 2.5-nearly 4 (he's October birthday) and is doing school nursery this year then will be reception age nearly 5 come September. We paid for the pre-school from 2.5-3, then it was funded from Jan-July (2 terms) before school nursery kicked in.
i think it depends on the nursery, whether a school one, community one, how many intakes they have etc? i'm not in hertfordshire but my ds is also june birthday and the school nurseries told me he wouldnt be considered till the jan intake but a community one would take him from sept, as they are entitled to their free hours from the term after 3rd birthday