We kept DS (ASD) in private
The problems we had (inexperienced staff, no knowledge of autism, poor outreach, insufficient SALT) would have occurred in both.
The funding was the same FT 1:1 except the private nursery got all the money from the LA when a school would have had to put it in from a delegated budget and I suspect would have been less likely to give DS dedicated 1:1.
The LA were desperate for him to move to a school nursery so they would not have to fund it as extra and could control what we did (we were doing ABA (private therapy) at home and asking LA to pay for this in longer term and private nursery supported it was really helping DS). The LA put huge pressure on the private nursery to say it was not working / not back us at tribunal. They really bullied the nursery. Despite this the nursery did help us and say the outreach was crap, the support insufficient etc and helped us win specialist provision for school age and even came as a witness for us at tribunal. They also allowed our private therapist into nursery to do training which a school would have blocked.
A school nursery would not have backed us on specialist placement - LAs make schools say they can take children whose needs they clearly cannot meet. The LA wrote to several school nurseries behind our backs on the basis if there was a teacher all DS problems would be solved so cheaply - every single school said it could meet his needs even though the Ed Psych said he should not be in mainstream. We felt as they were all employed and funded by the LA, school nurseries were too scared to go against the LA. They would not have been witnesses against the LA.
In theory all the support eg SALT etc should be identical in both. LAs should treat any nursery that provides free places (from age 3) in the same way as a school nursery and put in the same level of training, funding etc.
If your nursery will do the training, do the therapy and spend the money it gets on your child then stay put. Lots of schools will promise one thing but then do another or share 1:1 around.
Some schools locally are known to be great with SEN. Some are absolutely awful. One we approached put us off and clearly did not want him
DS was used to his nursery and they adored him, the staff had been through his diagnosis with us. We had ups and downs with them when the pressure was really cooking with the LA, but in the end they put DS first and I do not think a school would have done that.
A private nursery is less in the pocket of the LA and that worked well for us.
Also the hours are better eg DS could go for 4 hours not 3 and that meant the difference between me being able to keep my job and losing it.
None of my children went into school with their friends from nursery (DS3 has no friends anyway as his social skills are so far behind). It has made no difference, in fact often I feel sorry for the children whose parents insist they stay together as often they don't feel able to branch out / make new friends and lets face it how many of us still have the same best friend we had at 2?
If you have a good relationship with your current nursery that is worth a lot in my book.