I would expect a nanny to be able to alter her behaviour management strategies as the children grow, but if she's never looked after older children it might not come easily to her. However, I see encouraging good manners and good behaviour in children as part of my "job description" and if I was looking after children who were running rings around me I would be doing every thing I could to change matters, not just sitting there and letting it happen.
A really good book i would recommend is 123 magic (can't remember off hand who wrote it but it's widely available). Maybe you could bring it into conversation that, "as a family, you are going to look at some different discipline strategies, would she like to have a look at this book and let you know her thoughts?"
With regards to the mornings and getting ready for school; one thought I had is that maybe the children are behaving differently because they know you are around? I know in my own job, that if mum and dad leave for work when I get there in the morning, I can get the children washed, dressed, breakfasted and to school on time with no fuss or messing about.
When mum or dad stay at home a bit later to "help", I know that there will be messing around, dawdling, arguments and I can guarantee we will be late for school.
As a nanny, I see myself as firm but fair, I nip bad behaviour in the bud and the children I look after know the boundaries and the behaviour I expect from them. But, I know that the way I discipline them does change a tiny bit when I know a parent is in earshot - I suppose I worry about coming across too authoritarian (which is daft really, as the children are very happy and well behaved, so what I do is working), so for example, I might take a softer, more "talking through our feelings" approach, rather than the "ok kids, enough is enough" approach that I would use when in sole charge.
I know that, with me, this is a confidence issue, maybe something similar is going on with your nanny? (or maybe not, as she is saying they are also misbehaving when you are not there)
Sorry for the rambling post, it probably hasn't answered your questions but just wanted to put forward a few of my thoughts I had when I read your post.