For the purposes of commercial air traffic operating in a terminal/approach space area that has temperature compensation applied to waypoints in the active flight plan, if ATC assigns an altitude, will the assigned altitude be a temperature compensated altitude?
If so, how does ATC know what factor to apply to the procedural altitude?
If not, then an ATC assigned altitude (non-compensated) could be interspersed with compensated altitudes and possibly cause non-monotonic descent or ascent.
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 traffic.
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: ATC'er.