Given what you describe about the interactions when you are not present this sounds more like a problem you and the nanny need to work together to solve, around transitions and your presence. This doesn't sound like a problem with your child's relationship with the nanny.
Is your nanny experienced? If so, ask her opinions and thoughts, and try to be really open to feedback like "It would be easier if you left more quickly and with less attention to him. If you don't say a big goodbye and are just really matter of fact he will learn to be calmer about it. He's getting a lot of extra attention from you with this behavior - when you leave he's fine very quickly. So it's about attention seeking and we are feeding the dynamic when you stay in response to his cries." Etc...
That's what it sounds like to me OP - a really common challenge by the way, and not anything your nanny is doing wrong. It's just a different approach to leaving him that you need to adjust to - for everyone's benefit. It can run counter to your instincts, and you are three weeks in now so will have to reset this pattern and behavior, but I would bet that's all that's going on.
FWIW, I just went through this today w/ my five year old - so it's a really common dynamic. He was in tears, hanging on to me, not wanting me to leave for work, etc... So then I ended up in tears, had to just peel myself away and I beat myself up all the way to work. I arrived to an email from the nanny saying that he was totally fine - happily having breakfast and getting ready for school and was fine within a minute or two of me leaving. I got suckered!
So I would focus on rethinking your morning routine and doing that collaboratively with your nanny.
Good luck!