Hello,
when our old nanny (also EU, non-UK citizen) needed to open a bank account she hadn't received her NI number yet. I went with her to the local branch of our bank and they opened an account for her very quickly; I don't think they even wanted to see her contract! so you may wish to ask your bank in the first instance.
I am not sure of the position of au pairs vs NI number; technically as they have low income they shouldn't be paying NI, but there may be other reasons, which I am not aware of, for them to have one.
Anyway, if she is required (or entitled) to have one there are some useful instructions here:
https://www.gov.uk/national-insurance/overview
She will have to book an interview with the local Jobcentre; upon booking they will give her a list of documents she needs to bring. When I went through it the process was a bit tedious but simple.
Good luck!