If you write/email your local council they will send a full list of all the local providers including childminders and private day nurseries who are listed for the 15 hr provision. You will be able to register (often need to do it early) for a day nursery for the term following the 3rd birthday (so January for A) and do 2 terms there whilst also registering for your chosen state nursery for the following sept (3 terms).
I was a bit lost and unsure how I felt about it when I first started looking but honestly now I feel like it is a better option to do a term or two in a private day nursery. The one I have registered DS at is great, a much smaller group of kids (roughly 16 kids to 2 staff, if the group reaches 20 kids they get another staff member in compared to the state nursery which has 32 kids in a group and 2 staff) and is much more 'homely', to me it looks and feels like a much kinder/gentler introduction to 'school'. The state nursery is far more class rooms and corridors iykwim where as the day nursery looks much more like play group. I'm actually really happy that he will have a term to transition and get used to being left alone before he goes into a larger group and a more 'school' type environment.
One of the most popular here has the places filled about 18-24 months in advance and is unlikely to take January starters, they will take the child early, at September with the parents paying for the provision for that term which then secures the place.
Where I have registered DS they will start at any point in the year so long as when you register they have not yet filled their spaces, so I put his name down around April time for a start next April and am guaranteed a place now. I even considered leaving him there for his full 4 terms because it seems to much nicer, so much more what kids should be doing (playing and learning to socialise, learning to help themselves to lunch, pour drinks, go to the toilet etc), I still might, although my hope would be that he went to the state nursery and then got a place at that school, so would have made friends to start reception with and be used to the environment etc.