Answering for the US. Controllers in the US do not assign altitudes differently based on temperature.
From the AIM:
5−1−17. Cold Temperature Operations Pilots should begin planning for
operating into airports with cold temperatures during the preflight
planning phase. Instrument approach charts will contain a snowflake
symbol and a temperature when cold temperature correction must be
applied. Pilots operating into airports requiring cold temperature
corrections should request the lowest forecast temperature at the
airport for departure and arrival times. If the temperature is
forecast to be at or below any published cold temperature restriction,
calculate an altitude correction for the appropriate segment(s) and/or
review procedures for operating automatic cold temperature
compensating systems, as applicable. The pilot is responsible to
calculate and apply the corrections to the affected segment(s) when
the actual reported temperature is at or below any published cold
temperature restriction, or pilots with automatic cold temperature
compensating systems must ensure the system is on and operating on
each designated segment. Advise ATC when intending to apply cold
temperature correction and of the amount of correction required on
initial contact (or as soon as possible) for the intermediate segment
and/or the published missed approach. This information is required for
ATC to provide aircraft appropriate vertical separation between known
All pilots in the same general area of airspace are issued a common altimeter setting, which ensures everyone is on the same reference point of pressure as it relates to the weather, but not the temperature, specifically. ATC separates using altitude or lateral primarily and has a separation standard for this reason, so that if someone is off by a few hundred feet, it doesn't cause a collision due to being off as a result of atmospheric conditions. If nobody is correcting their altitude and basing it off ATC assignments and a common altimeter, there really shouldn't be much deviation.
Source: former ATC'er.