Single private girls school mum here (in charity shop clothes, a 15 year old car and a leaking bathroom before anyone makes comments about the wealthy elite).
I definitely would not move your child up early, it really sets them up to fail. It is a huge advantage not being one of the youngest kids in the class.
I know a parent who put their September born daughter in our nursery for this very reason and it did her the world of good. We often have two or three kids a year who come just for nursery and go back to state school for reception and others who choose to stay.
If you can get them into a private school nursery for the year I would. Don't try to get them to jump a year. My experience locally is that the nursery staff work to set the children up for Reception and it showed with the children who had been at other nurseries.
Some private schools can accept the 15/30 hours free childcare and take childcare vouchers. Mine are at a GDST school (there are 26 across the country) and they take them. It is worth asking. One thing that may be an issue is private school nurseries tend to run on term time only and we have longer holidays. Finding childcare for the weeks when everyone else at school is a nightmare.
For the future, do have a big think about the finances when you are looking at reception onwards. Try not to get seduced before you have done a ten to fifteen year financial plan. The fees were easy to plan for, it's the uniform, trips etc, charity days etc that add up. Our second hand uniform shop is fantastic but if it wasn't I would be spending £750 per child per year on uniform alone. Before the vat, school fees locally were increasing roughly 4% year on year and there is always a huge financial jump to senior school. Also factor into your budget for a cost of living increase too (ie what happens if your 4% mortgage became 8% and do you want to move house because this chunk of cash coming out each month will affect how big a mortgage you can get).
I don't regret a penny and I have skipped meals and the heating to make the finances work, and our annual holiday is three days camping in August (and yes I know I'm lucky to be able to). But the costs were definitely more than I carefully budgeted for and it is a challenge.
So in short, take advantage of a good private school nursery for this year, definitely apply for mainstream school places in the meantime and have a big think about what you want life to look like.