I think a nanny cam is worthless, for technical as well as social/legal reasons. In a sentence, it cannot replace trust. Let me first answer your questions, then provide some thoughts:
The video on the main page is enough to instill fear in me
I think the video you link to is made explicitly to put the force of fear in you, with "you" being a potential customer, and "customer" meaning profit for the shareholders.
--> This is not about the safety of your child.
Could it be termed illegal? Should the presence of a camera be disclosed?
Yes, in some places it certainly is illegal, and in some places you must disclose the fact in advance.
--> Check your local laws.
Surveillance cannot replace trust
If you trust the nanny so little that surveillance is necessary, then you should question your choice of nanny. Of course it can be very difficult to find someone to put that much trust into, and even if there's trust, it's no guarantee against bad events -- accidental or not.
Technological limitations make the whole exercise questionable
Bad events might take place inside the camera's viewing angle, or off-camera. Because one camera cannot cover the entire home, you'd need several just to cover the most obvious places. This seems impractical and expensive.
In many places in the world, video surveillance is only allowed (legal) when the observed people are informed in advance. If you do inform the nanny, she might deliberately move off-camera. If you illegally monitor but don't inform, then using that evidence might get you more trouble than it's worth.
Monitoring provides no protection by itself, it merely tells a story after the fact. If you're watching an event happen, what can you do? You can't assist directly. Sometimes calling your home phone (or even emergency services) might be helpful but lacking any statistics I suspect that case to be very rare.
I was going to write that the only situation in which I think a nanny cam might be useful is when using it as a video baby monitor -- say, when you're sitting downstairs while the baby is supposed to be sleeping upstairs. Having a (night-vision) cam would save you the trip of going upstairs to check. But that's a video baby monitor, not a nanny cam, and if it's that difficult to reach your baby then you might consider being closer to it in the first place.