If the nut is simply detached it will continue to orbit the Earth along with the satellite since it already has velocity equal to the orbiting satellite. This is how satellites are placed in orbit in the first place. They are taken to the required elevation and accelerated to attain the required speed and simply let loose.
If the nut is ejected with force against the direction of orbit the nut will revolve around the Earth in a decaying orbit and will eventually fall to the Earth. It will most probably vaporize on entry into the Earth's atmosphere and will never make it intact.
To make it land in one piece, it needs heat shields to prevent it from burning up due to the friction with the atmosphere on re-entry. However I don't think you can heat shield something as small as a nut