Not giving anything away by saying that if there was a fire in the building, but not in the Ops room, the Ops rooms have a slight positive pressure compared to the rest of the building and also have measures to make it safe (allegedly) for operational staff to stay in situ whilst the building burns around them for quite a long time... longer than I'd want to stay there, and longer than it would take to transfer every aircraft to other sectors outside of the Swanwick AOR.
As for controlling aircraft procedurally and them all being level seperated - maybe more likely on airways (but still not 100%), but certanly not in the TMA - it would be carnage if it (total failure) happened when it was anywhere near busy, TCAS might save the day.
However it is not a concern because instantaneous failure through loss of power or complete loss of radar displays is almost impossible... there are of course other ways that a facility could be brought to a complete standstill instantaneously, but if that happened the controllers and staff would not be around or in a fit state to do anythng.