You need to understand how TCAS works to get the whole picture. The TCAS units in the approaching aircraft "talk" to each other and determine who will climb and who will descend. Then they give the aural and visual commands in the planes. One will say climb and one descend and the event is over. All airline operator are trained to follow TCAS over ATC unless there is some extreme reason not to. ATC has no idea which plane is going to climb or descend since that issue was decided by the TCAS units in the planes. If TCAS says climb and ATC say descend, you climb - because the other plane is going to descend.