Depends on the situation. TCAS instructions are sometimes issued when ATC has already ensured separation. I saw this one time. Aircraft A climbing to ten thousand, Aircraft B descending to eleven thousand. Both aircraft got a TCAS alert, A to continue climb through B per TCAS instruction, B got TCAS alert to increase descent rate. They passed in the clouds and never saw each other, the targets merged on our RADAR. That's a full digital RADAR, and they were about 10 miles from the antenna. so it was VERY close.
In a situatiuon where ATC has blown it, usually the Controller in question is relieved by another Controller so he/she can participate in the investigation process.
In the US ARTCCs (ACCs) the computer will alert the supervisor if a Controller has less than required separartion, resulting in the above process as well. That device is not in our Terminal Control facilities.
That's the US answer. Hope someone from NATS chimes in as well.
I don't understand everything I don't know about this.