OP, I am sorry you are being ragged on. It is totally understandable that you would want your son to have a smooth transition to preschool and I think it's lovely that you understand and value the importance of your nanny in his life. And using a week of your own probably limited time off to handle that week yourself may not be plausible for you. I get that.
That said, you have to realize that this isn't something you can say no to easily. Sounds like this is her chance to see a close family member and that it's a rate opportunity for her and she has given you plenty of time to get creative with solutions.
You could enroll DC in a few weeks of camp so that that week isn't such a big transition. You could start him at school for the first 2 days (with you home to guide him) and have him stay home with a backup nanny the latter 3 days. You could delay his start at school to a different week. You could take the whole week off to help him transition. You could introduce a new sitter now by having her watch him every other week for a date night so that you and he feel comfortable with someone new for that week.
The good news is that even if you had to stick him with a complete stranger on the same week he starts preschool, he will likely be just fine! The transition to school seems so huge to you right now but most kids adjust within a month and a year into it, even a really rocky start just feels like a blip on the radar. So let your nanny go as she requested, do your best to find successful workarounds and know that it will all be fine. Really.