Roffa,
I am well aware of what EGLL Dir does at night, thanks.
They will usually have assumed the reclined position well before 0300 - if the police call, it does not exactly involve much in the way of monitoring.
The ATCO will still possibly be reclined, or chatting to the GS, but at the same time will keep a weather eye out on the on or 2 aircraft they have on frequency.
A/C that to be honest, have only called them because they have to; not because they need to to safely accomplish their task