PANS OPS Volume 1 ICAO DOC 8168 is the reference.
Here is a quote I posted earlier;
This chapter deals with altimeter corrections for pressure, temperature and, where appropriate, wind and
terrain effects. The pilot is responsible for these corrections, except when under radar vectoring. In that case, the radar
controller issues clearances such that the prescribed obstacle clearance will exist at all times, taking the cold
temperature correction into account.
My highlight.
The UK AIP is still awaiting the appropriate text to be inserted.
However, if you get a cold snap and the temperature at LL is -10, then the minimum safe altitudes have to be increased by the cold weather corrections.
I as a pilot will do it for own navigation flying, but if being vectored, then as the person responsible for terrain separation, the controller must make the corrections. I will be checking of course!
Regards,
DFC