Why does flushing a toilet or running a clothes washer sometimes change shower temperature?
This happens because water is diverted away from the shower to the toilet or washing machine.

What happens next will depend on the type of shower and it's age. Older, non thermostatic showers will be affected to a greater extent.

For an electric shower the flow of water through the shower will be reduced which will have the effect of heating it up further - though the thermostat should keep the effect to a minimum.

For a mixer shower it will depend on whether the cold water is reduced - in which case it will get hotter, or the hot water via the heater is reduced - in which case it will get colder. Again a thermostatic mixer should keep the effect to a minimum.

In any case you'll get a much reduced flow through the shower.

Power (pumped) showers shouldn't be so affected as the pump in the shower is controlling the flow rate.