How to move spheres in 3d without collision
Here's how I would do it:
Create an oriented bounding box around the path of each sphere, with dimensions equal to the sphere's diameter. Now perform collision detection on all of those bounding boxes. For the ones which do not collide, their correlating sphere's can travel at max velocity.
For the bounding boxes that DO collide, do they collide at a point equidistant to the spheres' starting points (+/-  radii)? If not then they are free to move at max velocity. 
If they DO collide at an equidistant point(+/- radii), then you must slow one sphere down so that it will not collide.

Hope this helps.
Good luck!