Is it reasonable to expect the Nanny to pay their Nanny insurance if I pay for the registration and the first aid course?
Yes. Though the nanny may already be registered and have first aid training. Given it can take time to register, you may want to look at candidates who are already registered, depending on your proposed start date.
What happens if they leave shortly after I pay for everything?
I am not sure. I suppose you could make repaying it should they leave within six months a condition in the contract, though not sure in reality you would want to go through the legal process to enforce such a thing, for the sake of what may be £200 or so. Small claims court costs and it's the time and hassle you need to consider.
Qualifications wise it is tricky. Education.gov.uk: Qualifications Finder - probably best to make sure it's on this list.
Would a student who had completed the first year of a degree in childcare be well enough qualified?
No, not sufficient in my view as until they have their degree they have no qualification. So look at what they had prior to starting the degree course.
There is a short Common Core course that can be done... OCN Common Core Skills which can be done by distance learning (cost £138). Would take a while though, I can't remember how long it took me... not long for the paperwork but took a while to get it marked.
Another candidate has completed "Foundation Child Protection course" I really have no idea what that means
Me neither, it could be anything. Possibly something like this which would not be suitable for Ofsted Registration. It's being aware of child protection issues, which is just one bit of the Ofsted requirements what anyone working with children should have training in on a regular basis (plus finding out local child protection procedures, as things can vary between LA areas).
Have you sorted out payroll side of things? Keep in mind the applicants most likely have another job, as the hours you are offering are low. So agree a Gross salary.