Oh my goodness @AdifferentGoat, the whole situation sounds overwhelming - you seem to be spinning lots of plates whilst trying to walk a tightrope in stilettos over a bear pit!
You are not being unreasonable, you need to feel comfortable in your own home and your Nanny is not doing her part to make this work. You have been very (too) kind, self deprecating and grateful thus far, and it sounds as if you have been the one making all the concessions towards a harmonious household, whilst your Nanny is unable to even accept simple instructions without some sort of passive aggressive response. Can you imagine acting this way in your new role? Your Boss asks you to complete a task for them and you avoid looking at them, roll your eyes and heave a dramatic sigh... How do you think that would be received? Her behaviour is not acceptable, please stop beating yourself up, it's not your fault that she seems unable to behave professionally (or even like an adult!)
I agree with PPs that you do need to replace her, but given the fact you are in a probationary period, I would hold fire until you have been confirmed in post, so that you can give the process of interviewing for a new Nanny the time and energy it deserves.
I would suggest you write a bullet point list of the issues to address and have a weekly catch up where you discuss progress towards these 'objectives' (line manage her as you would an employee in your working life), if she improves and you're happy, then hooray! If not, you can trundle along for a couple of months until you pass your probationary period, safe in the knowledge that there is light at the end of the tunnel (although I suppose it is possible that she might start to take your feedback on board and begin to behave appropriately, which would save a whole lot of drama all round - I'll keep my fingers crossed, but won't count any chickens!)
However you decide to approach things, I sincerely hope that it won't be too long before you are feeling comfortable in your own home, succeeding in your new role, with a happy, healthy, contended baby handed over to you when you finish your working day.
And I hope your DH stops being such a knobber!