Granted I do not use tapwater, I use bottled water for my aquariums due to absolutely abysmal tapwater quality.
That said, I put enough packs of water into the lounge (where both aquariums are) the day before water change day. I do not use any heating in my flat and the lounge never goes below 18 degrees even in the harshest of cold nights.
When doing water changes...in summer when normally the aquarium heater is off, I turn it on for the duration of the water change and for about an hour afterwards...in winter the heater is left on 24/7 anyway.
Since I use bottles of water I can limit the flow of the water coming out. So I dribble the water from the bottles, moving the bottle in a figure 8 around the aquarium so that the new water isn't concentrated into one area thus causing a potential colder spot.
The figure 8 method and slow pour would work just as effectively with tap water whether the aquarium heater is on or off, ideally it should be left on so that it assista in balancing any temperature differences that might exist.
Slow pour, figure 8...not had a single bad effect such as chill shocked fish in decades doing it that way.