Don't know if you've seen this OP?www.autism.org.uk/about/diagnosis/children.aspx
NHS diagnosis waiting lists are 18 months to 2 years in many areas. You need to make a case and see the GP with your evidence in order to obtain a referral for an NHS diagnosis. It's worthwhile getting into the system and on the list as well as pursuing a private diagnosis. IF the private diagnosis is recognised, you can drop off the list, if it's not, you should be able to obtain some help from school whilst you're waiting.
It's a horrible system, we all know interventions at school should be based on individual needs, there should be no requirement for anything else, but in practise a diagnosis can make some schools provide services they would not do without one. Some schools are great at providing interventions, some are just rubbish.
Re private diagnosis, SOME professionals who diagnose privately also work for the NHS part-time or are recognised by the NHS.
That includes the Lorna Wing Centre and Daphne Keen.
Other private diagnosis available here you will have to ask if they will be recognised by the LA/NHS. www.autism.org.uk/directory/browse/cid=80~aid=1.aspx
Are you sure you're not mistaking aggression for overwhelm - that your DGD cannot cope with a situation therefore lashes out as a defence mechanism? Aggression is NOT a marker for autism, sensory overload and a reaction to that can often be fight or flight, often observed as meltdown or shutdown.
It's common for a child with ASD to hold it together in school (it's called Masking) so school "see nothing" then the child either melts down or shuts down on leaving school, on the way home, at home etc.
Try MN's boards SNChat or SNChildren.
In short, do not listen to anyone like a GP or SENCO or teacher who says your DGD does not have autism because e..g. they make eye contact, they have empathy and all the usual 'can't be autistic because' myths. Those people are not qualified to make that decision. Make lists of how your DGD's behaviour is different to her peers particularly in communication and sensory overload, see the GP for referral and book a private diagnosis as well.