Does your DD have a diagnosis, has she been assessed by anyone? I think an appointment with an educational psychologist would be a good idea as an EP would be able to advise regarding a suitable school placement. A SALT who knows the school should be able to help as well.
You have to remember that an independent school does not have to accommodate any special educational needs. Any support your DD needs you will have to arrange yourself, and you have to hope that the school will co-operate. The school can turn around at any point and decide that it is 'no longer appropriate' for your DD and kick her out. Sorry to sound harsh but I have heard this happen, and it happened to us (although my DS has ASD so more needs than your DD). If the school is very academic my guess is that they really don't want to deal with anything beyond mild dyslexia, but I'd love to be proven wrong.
However, while she is in nursery, IF the nursery is in receipt of the Early Years Grant, she is entitled to all the services from the Local Authority AND, more importantly, the nursery is bound by the SEN Code of Practice which sets out how the nursery must assess, monitor and support any special educational needs, however mild or severe. They should have a SENCO and an Early Years support officer. You should go chat to them and about your DDs needs and also to gauge a) how clued up they are about special educational needs and b) how supportive the school is likely to be.
A state school has to provide an education appropriate for your DDs needs, this is their statutory obligation. Independent schools have no such obligation. In a state school your DD can access all the available support, and in our experience it makes such a difference dealing with teaching staff who both understand SEN and the support network and want to support pupils with SEN.