If she has an identified speech disorder perhaps she should be statemented, this usually starts at 3.5.
I have 2 DC with (varying) SN in the private sector. My DS' SALT goes into his school, as does the cluster area senco, physio etc so do not understand why the SALT will not go into the school.
I'm going to be really honest here, but you need to consider what is right for your DD.
My DS joined my DD's pre-prep's nursery - his name had gone down before birth - but I knew that a highly academic/push environment was not right for him. After a few weeks it became obvious that he was not welcome and not wanted he did not fit their criteia. Hewas still in nappies, did speak, I had to pay for a 1 to 1 and after only 4 weeks they insisted he be kept back from his peer group the next year.
I found another private school that is smaller and more nurturing, they love him and value him as a pupil. They bend over backwards to accomodate his needs and make him welcome.
The difference in one year has been astounding, they have completely understood what motivates him and how to help him achieve.
At his last school I was told his pencil skills were poor. At his new school I was told they would be using trays of sand and salt to help him understand 'mark making', he was given buckets of water and big paintbrushes and he painted on the dry playground etc all these things were done directly to assist his understanding of prewriting skills - I was never once told his pencil skills were poor (which they are).
Just because you are paying for the education doesn't mean it is the best one for your DD.