If you have the buzzing when you are depressing the string on a fret, it almost certainly has nothing to do with your nut. Nut problems commonly show themselves when you hear a buzz plucking an open string that you do not hear when you pluck the string on a fret.
As far as your pickup moving: As others have said, it is hard to say without seeing a picture of your guitar, but both of my electrics have a cover over each pickup (black plastic on one and brassy looking metal on the other). The ones over the humbuckers are much wider than the ones over the single coils, and these can be easily adjusted with some small screws that are easy to see if you really look but hard to see from any distance. If you unscrew those, you can take the cover off and see that there is a spring system underneath there that keeps the cover in place after you have adjusted the height of the pickup/cover, but since it is on a spring, I can push it down a little and it will pop back up. In other words, the fact that you can mush the "pickup" around a little is nothing unusual if you have pickup covers like mine.
Again, not knowing what you have: you said that the buzz was from the neck pickup, I would suggest that if you have the aforementioned pickup covers on your guitar (they are very common), that you try lowering the cover over whatever pickup is closest to your neck. On the single coils on one of my guitars, that will mean actually lowering the pickup a little. If you are not tech saavy, I would suggest lowering it one half turn on each screw (there should be one on each side), that way if you later feel you have messed something up, it will be easy to put it back the way it was. See if that works, or makes it better but still buzzing. If the latter, try another half turn and see if that does it.