I don't feel that a nanny caring for one child has a lot of responsibility - sure they have some, but most nannies work for families with 3 or more children (for example, I care for 3 under 5's plus an older school child), so far more responsibility.
Your nanny has sole charge of your baby - so should go along with any of your wishes, subject to them being legal and reasonable given the agreed contract. You are the Employer, YOU call the shots.
YOU decide when your child will be weaned. Your nanny can advise you, as can your GP/Health Visitor, but ultimately it is your decision.
6 month old I care for, stated on solids at 4 months - parents choice.
Booking Classes - again this is your decision, as you are paying for them. Nanny can tell you what classes are on, and why they feel those classes are suitable. Personally, I tend to do PAYG classes - as then we only pay for what we attend.
Routines - established routine should be followed, and discussion with parents with regard to changes (routines do need to change as children get older).
You will have to have a formal chat - written warning stage. Follow any disciplinary procedure which was outlined in the contract.
ACAS - Self Help Guide to Producing Disciplinary Procedures may be of help, if the contract does not already contain discipline procedure.
Make it clear to your employee, that she is your employee and she does what you say. If she doesn't like that, she can leave. If she continues not to do as you say, then she will be sacked.
Caring for one child seems a great job to me... alas I'm not keen on working in London.
Good luck with your chat with your nanny.