Nanny teething problems ..... Help Please!

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I recently returned back to work after my 2nd DD. We have not had an easy time sorting out a nanny. Our initial nanny didn't have any paperwork so we didn't ask her to start. We found someone else but she resigned after 2 weeks because she didn't want to do the childrens washing(!) - she was lovely and cuddley with the girls and both of them dd1 - aged 3 and dd2 almost 1 seemed to really like her.

We have found someone else who my husband and I really like (she stayed with her previous family for 5 years, and we really liked that as we'd like the same level of continuity) but she is a lot stricter than the previous nanny, and my eldest is now whispering to me as soon as she sees her " I don't like x, she can't come to my party" and she demands that either my husband or I take her to nursery and has a screaming fit if we leave her with the nanny. I should also mention its only week 2 of new nanny.

What should I do? Is this just teething problems or should we consider getting another nanny?

I think you just need to have a chat with her,
we have got a lovely nanny who has been with us for over 3 years. At the beginning my older DD's would say that the nanny wouldn't let them do certain things.
Just silly things like if they were having a snack then they didn't have to sit at the table, and were allowed to watch a certain programme after lunch. But they were things that we nearly always did, so DD's couldn't understand why they weren't allowed to do it.
But from the nannies point-of-view she has got 2 girls whinging about having to eat a snack at the table and saying that 'mummy always lets us watch X programme after lunch' - understandably probably thinking that they were trying to pull the wool over nannies eyes! I just had a chat with our nanny and we sorted it out - I explained to my DD's that when they are with the nanny then she is in charge.
From your nannies' POV your DD is saying 'mummy always lets me have a chocolate coin after music' (which is true!) but maybe nanny thinks that your DD is just after a bit of chocolate!

She sounds to me like she is just trying too hard. Maybe fairly new nanny or worked for families with 'no tv rule' before. I was a bit like this when i was very young and a nanny.
I think you just need to sit down with her and explain what your expectations are.