Lateral separation is a procedural tool and the position where there is one mile between the possible tracks of two aircraft after applying navigation tolerances. E,g. With 40 degrees between two VOR radials, the lateral separation point along those radials is about eight miles from the aid.
In a double go around, the ADC will use visual separation until a radar standard is established. After that, the aircraft will be transferred to the TCU.
There is a complete section in MATS covering visual separation, how and when to apply it. ATCs are trained on double go arounds in the simulator. They are rare but easily managed despite the public's and Nick Xenophon's perception.