Missy comes from a race of time travelers, and you're wondering how she got ahold of people from different points in history?
I think it's important to note that they said the Nethersphere was actually a miniature copy of the Matrix. Knowing what the Matrix is explains quite a bit about what's happening...
The Matrix was Gallifrey's main computer system. It monitored the whole of time and space, and saved any information it decided was important. All Time Lords and students had limited access to the Matrix like a library system, but higher ups in the government had access to its higher functions. The Lord/Lady President had full unrestricted access.
The Matrix was also capable of storing copies of people's minds. For example, every Lord/Lady President in Time Lord history had a copy of him/herself inside the Matrix.
Here's the thing: You could also physically walk into the Matrix and interact with anybody/anything inside. The season finale back in 1986, for example, took place almost entirely in the Matrix-- the Doctor, the Valeyard, and Mel walked through a Matrix Door and had their final confrontation inside a fully interactive landscape.
Missy's Nethersphere is a machine capable of looking at the whole of time & space, uploading minds, and can be physically interacted with.