I have a nanny but she doesn't have children as yet. She has had to take emergency leave due to a family member's sudden illness which is more or less the same thing.
I would try to view it as an employer. If you have to take time off work because your children are ill and you don't have cover (which happens even with a nanny) then you take it as emergency family leave or holiday depending on your employers policies. If you call in sick yourself you put yourself at risk of being first in the line for redundancy but people do, do it.
You should ask what arrangements she has in place for emergency care for her child(ren) and on what basis that might fail.
You should speak to your payroll agency if they offer contractual advice
Look at the legal right for Parental Leave - your own office will have a policy. I don't know if you can establish in advance if she has a disabled child which would mean a weeks notice rather than 21 days. It could be a discriminatory question.
Attached is an excerpt from our policy. It doesn't state on what grounds your employment could be terminated for a poor attendanc on the basis of repeat emergency leave requirements. You could perhaps insist on a period of parental leave but I am not a HR person.
Emergency Leave for Dependents
2 Length of time off
This policy entitles you to reasonable unpaid time off which will normally be one or two days. If you require more than two days, please speak to your Manager. Please note that, dependent upon the circumstances, you may be asked to take any further leave from your holiday entitlement or as unpaid leave (as per the Unpaid Leave Policy).
Whilst there is no restriction on the number of occasions that time off for dependants may be granted, where amounts of leave are deemed as excessive you may be asked to meet with your Manager to consider what alternative arrangements can be made.
3 Pay
Please be aware that any time taken under this policy will be unpaid.
4 Notifying your Manager
In order to be eligible for time off under this policy you should, as soon as reasonably practicable, and where possible in advance of your start time, notify your Manager or HR representative of your absence, the reason for your absence and how long you expect to be away from work.
If the emergency occurs when you are at home, please make sure that you phone your Manager as soon as possible.
If you need to leave work at short notice, you are required to agree this with your Manager.
Please be aware that your Manager may ask to see evidence to support the reason you require time off for dependants (such as a doctor's note, or letter from the school).