The two particles are initially at rest, so the initial velocity of their centre of mass is zero. They move towards each other due to mutual attraction, which is an internal force. Since no external force acts on the two-particle system ($F_{ext} = 0$), the acceleration of the centre of mass is zero ($A_{cm} = 0$). According to the law of conservation of momentum, if no external force acts on a system, the velocity of its centre of mass remains constant. Since it was initially at rest, its speed will remain $0$ at any instant.