The question as originally stated contained an error. If two objects are mutually attracted starting from rest, their velocities cannot be in the same direction. Though the problem did use the word "speed" which is usually equivalent to the magnitude of the velocity, the question asks about the velocity of the center of mass. This really must have a direction associated with it. I changed the question so that object B has a negative velocity.
Since no external forces act on the system, the center of mass keeps moving just as it did before the two objects started to exert forces on each other. Since the objects were at rest, they center of mass will remain at the same place. If the two objects are simple spheres or can be thought of as point-like, they will collide at their center of mass.
This is an example of conservation of momentum. Since the system had zero net momentum to begin with and no external forces act on the system, the momentum must be zero at the end.
Knowing that the forces on each object must be equal and opposite, we can deduce from their velocities the relation between their masses. Object A must have twice the mass of Object B.