Quick Answer — How Much Does Washing in Cold Water Save?
Nearly all the energy a washing machine uses goes into heating water. The motor, pump and controls draw very little. So the saving from washing cold is the heating you avoid, and that follows straight from physics:
Energy (kWh) = litres × temperature rise (°C) × 4.186 ÷ 3,600 ÷ heater efficiency
The 4.186 is the specific heat of water in kilojoules per kilogram per degree, 3,600 converts kilojoules to kilowatt-hours, and one litre of water weighs one kilogram.
A 57-litre load heated from a 15 °C inlet to a 49 °C warm wash is a 34-degree rise. That is 57 × 34 × 4.186 ÷ 3,600 = 2.25 kWh of heat, or 2.37 kWh through a 95%-efficient electric heater. Add 0.09 kWh for the machine itself and a hot load costs 2.46 kWh.
The same load washed at 20 °C needs only a 5-degree rise: 0.35 kWh of heating plus 0.09 for the machine, or 0.44 kWh total.
The difference is 2.02 kWh per load, and 96.3% of a hot load is water heating. At 17 cents a kWh that is 34 cents saved per load.