Thanks everyone.
Just to clarify I don't record her because I am trying to document evidence of her knowledge but as part of the memories for the family as we all grow older. As an example, she might be pretending to be a teacher and I record her (sometimes she's aware I'm recording and other times she's not). She holds up a flashcard as if teaching a class, she says the letter sound and asks her audience to repeat what she's said and we might end up playing snap afterwards with the recording still on etc.
What I wanted to demonstrate by showing the teacher is not really her academic abilities but that she is neither shy nor quiet at home and to borrow Stealthsquiggle's words, she is a dominant bossy boots, confident, loud, free spirit. I have shown the teacher the video and she was quite surprised she could speak so confidently and at what she knows. She actually says she was doing exactly what they would do one on one in school and they are going to get her to be responsible for some aspects of their play time which will encourage her to initiate play, speak out loud etc. I only supplement what she's already learnt in nursery in whatever I teach her at home. It's a Montessori school nursery not a childcare nursery if it makes a difference, so they are teaching them though it's all through play. I believe she is expected to know these things but if she doesn't talk to the teacher, there's no way the teacher would know that she does. And from my earlier meeting with the teacher, I don't think she did.
And whilst I appreciate it's only nursery, she's been there for 2+ years and she obviously is not confident enough to be herself. She does like the nursery as she is always eager to go, talking about what she will do and how much fun she will have with her friends. But somehow by the time she gets in the Nursery building, the eager child has become extremely quiet. It's like a switch. Mostly at the nursery, she only talks when she is spoken to even amongst her friends there and same applies to her younger sister when their classes are mixed in the afternoons. Her sister on the other hand just started in September, is confident and ensures everyone knows she's around. I know every child is different. I probably wouldn't worry as much if she was the same way at home because then I'd say it's her nature.
If she is extremely quiet and shy in school I don't see how it won't impact her academically though IMO she is bright and always eager to learn. I will try to stop worrying but keep looking at ways to encourage her still.
I apologise, it's ended up being long again.