Deep breath! I had the same panic, it made no sense to me either 
Private nurseries have different rules about the funding, but most will take it. There are lots of different ways that they take the 15 hours term time funding off your bill, so you do need to talk to them about it. Ours seems to not want the hassle of working it out so they just average it out over the year and take the average off your monthly bill. Other private nurseries might charge you £X minus the funding during the term-time, but then £X during the holidays when there is funding.
Some 'school' nurseries also offer what's called wraparound care. This can vary a lot. One of the places near us offers wraparound care during school hours, which isn't enormously helpful. Another place offers wraparound care from 8-6 - essentially the same as a full-time private nursery.
It can get quite confusing so I would do a little research now into the local 'school' nurseries and then check their website or email them to find out if they offer wraparound care too. However, don't say anything to your private nursery until you get your places at school and wraparound care as sometimes a few things need to fall into place at the same time.
For example, our nearest school nursery offers two 3-hour sessions every day, one in the morning and one in the afternoon. These are the free, funded hours. They also offer private wraparound care on the same premises from 8-6. There are (annoyingly) two different things. So you had to apply to the council and hope to get allotted the morning slot at the school nursery. Then you had to see if the private wraparound care had space for your DC from 8-9am, and from 12-6pm.
Did my head in - and we cocked up on the application (as in, they'd already filled the waiting list for the private wraparound care places before the school nursery had even held it's open days
).
Essentially, though, it would be rare for your private nursery not to take the funding, and I imagine you'll be able to just carry on with your full-time place with some decrease in your fees. Do remember to ask them what they do in the 'pre-school' room though. There should be some move to 3-4 year olds being introduced to a little bit of education, with some phonics and writing and numbers. I wouldn't worry too much about your DC not going to the nursery linked to the school - loads of children won't, it won't make a difference when they all start in reception. What's more important is good care at a place everyone is happy with.
You'll get letters from the council telling you when to start applying for places, but your local council website will tell you the dates too.