I'm a nanny with lots of experience in this field and I do understand how hard it is for families to find the right nanny! It's also nerve wracking for nannies to find the right family too!
All of the above checks are very important indeed although gut feeling is crucial! They (candidates) may have all relevant documents in check, DBS, references and so forth but...if it doesn't feel right you'll know!
All the families I've been working with (very important to be a team!) have checked my references after or before interview. None of them asked for DBS check...and when years later I asked how come I was told that they knew I was the right one also I knew they were the right family for me! So checks are important of course though you should go with your instincts! Btw, I always suggest a trial of a week to see how both parties get on.
Best of luck!