What you’ve described isn’t screaming ADHD to me, but it is quite suggestive of autism, but who knows. She could get a diagnosis of one, both or neither.
The school’s SENCO will often observe how children are getting on to see if extra support is needed. The school assessment is most likely purely this. They definitely won’t be diagnosing anything merely seeing if more intervention/support is required. From there is might be more adults in the classroom, specific interventions/small group work, or they may go on to ask specialist teachers or educational psychologists to advise the school on how best to support. (This is all idealised. Often this doesn’t happen because the school doesn’t do what it should, funds don’t allow or there are long waiting lists for support from the LA.)
It differs in different places but the school won’t necessarily refer on for diagnostic assessment. This often involves going to the GP and getting a referral from there. Look up how it works in their area or speak to the school. The school, rightly so, will be focusing on identifying and meeting needs not naming conditions.
If DGD needs aren’t minimal, and it sounds like they aren’t, then either the school or your DD can apply for assessments to be carried out with a view to getting an EHCP. This outlines DGDs needs and what has to be provided to meet those needs. It is legally binding, so if, for example DGD is deemed to need 1:1 support for maths, if another child comes along who also needs enhanced support the school can’t just lump that child in with your DD and make it 2:1, like they could if they were just managing resources in the best way without anyone having an EHCP.
The actual diagnosis should make no difference, provision at school is put in place to meet identified needs, but in reality having a diagnosis can make some difference, although probably less so that it used to as the waiting lists for diagnostic assessments are so long that there are so many eg autistic kids out there without diagnoses. If DGD might need a special school in the future sometimes they state that they only take autistic children and so a diagnosis would be needed for that. Also short breaks provided by the LA are sometimes specific in that they say autistic children are eligible, but at other times they include, in addition, those with social communication needs or those that at waiting for assessments.
Diagnosis is often a bit of an anti-climax in that you think having the diagnosis immediately results in people swooping in to help. In reality nothing changes really other than you are a bit surer about the issues that a child is facing as someone else has confirmed they see what you see. Most of what you can do after diagnosis you can do before - read about autism and ADHD (Cerebra have a library of books you can borrow for free on such subjects), engage with parent carer organisations, put in place strategies to help - visuals, predictability, be aware of sensory overwhelm, put in place equipment and routines or help with sensory regulation, specifically teach social skills through social stories etc,. Most of all though just tune in to her and see what she needs and wants. Parent (or grandparent) the child you have not the child you would like to have had or the child you would like her to be.