Call HMRC and ask to speak to a Status Officer - not the Helpline, actually get to speak with a Status Officer as the situation can be confusing and you need expert advice.
Depending on your contract, you may be able to be self employed - but generally speaking it is unusual for a nanny to qualify for that status... as a nanny usually does not have sufficient control over working hours, working days, duties etc.
However you don't mention set hours, thus depending on what is written in the contract, you never know HMRC may say that you could do it on a SE basis.
It is advisable to have Insurance, though it is not currently a requirement under any legislation I am aware of. However, if the parents are paying using Childcare Vouchers or Tax Credits, then you will need to register with SureStart CAS or Ofsted Childcare Register, to be able to accept those, and it is possible that in the future, having insurance will be part of the requirements for that scheme (though is not a requirement at this stage, for CAS).
Trial period could be for as long as your employer likes - it should be mentioned in the contract. Typically 2 weeks to 2 months I would say as a ballpark figure.
What is your contract situation... do you have a written contract of any kind? This could in my view include any e-mail which says what your hours / duties are to be.