Sounds to me as though perhaps she is doing some kind of childcare course aimed at nursery workers, rather than workers in a home setting - not sure though, hard to tell... knowing course title would help.
Liking a potential carer is always a big plus point. Seeing how they react with your child and feeling positive after that event is also a major plus point.
No formal training isn't always a negative thing, after all, as a parent what training do you have for raising a child? As she is already on some kind of course, that shows some commitment to doing training... though I'm not sure the course she is doing is for Childminding specifically.
Until she is a Registered Childminder she won't be able to have Childminder Insurance. If she works at your home as a nanny, then Morton Michel can provide Public Liability/nanny insurance.
By coming to your home, she would be your employee. If the amount you are paying is over £84 a week (I think the amount is), you will need to register as an employer and make Tax/NI deductions.
Getting a CRB check done for a nanny can be tricky, though if she is registering as a childminder then she will get one done that way eventually. If undergoing Nanny Approval (you mention approval process, rather than registration process), then will get one that way as well. Another way to get one, is for her to register with a nanny agency and ask the agency to do an Enhanced Level CRB check (cost £40-£50 typically via a nanny agency).
If coming to your home as a nanny, then you need to consider what happens regarding her own child... will her daughter also come... did you DS get on with her daughter?
With regard to calling a Business on a Sunday (childminders are a business), I feel that if the Business answers the phone, then they should be prepared to answer a customers enquiry. Personally, when I don't want to answer the phone, I screen the calls letting the answerphone take the call and then responding back in my own time.
Perhaps it wasn't a convenient time to chat, if so they could have told you that and taken your name and number and arranged to call you back later.