"The paediatrician did suggest yesterday that she would probably not at this stage get a Statement.Possibly not until other interventions had been tried and had been shown not to work...."
Sigh, is this really what you have been told, its all a fudge honestly to stop parents like your good self applying.
The time for waiting is finished, now its time to kick butt!!.
Your DD is already being failed at school.
She has not even been put on an Individual Empty Promise (an alternative mumsnet name for the IEP). As your DD is quiet, non disruptive and acquiescent in class she has fallen through the cracks, that happens a lot too. However, the fact that she is very upset after leaving school is very telling - what is happening here is that she is either just about coping during the school day and bottles it all up. After school she takes out all her frustrations on you. This happens a lot in situations when children who have additional needs are not currently having these needs met. Many children who are on the spectrum do find school bloody hard particularly if they are not additionally supported with a Statement behind them.
An important difference between a Statement and something like School Action Plus is that a Statement is legally binding. School Action plus is not legally binding and thus can have very limited support attached to it. It truly is worth very little.
I would therefore now apply for a Statement to the LEA on her behalf. This is all very daunting but no-one else but you is better placed to fight her educational battles for her. Also no-one else is going to do it for you.
Hopefully the school will be happy to see the Autism outreach workers, although some schools do think that, "they know best" and become mistrustful of such outsiders.
You are truly the best placed person to enforce a change re her educational needs. You have far more power than school ever would when it comes to a Statement application. Do make that application asap, do not leave it unduly as the further up she goes through the system the harder it actually gets.
www.ipsea.org.uk is a great website with lots of information on it.
I would also look into applying for DLA as well and get outside help like Cerebra to help you with the mountainous and devious forms.