A lot of modern products use "switching" types of power supplies. Switching supplies regulate the voltage to the appliance by means of fast transistors. As long as the supply can handle 240 on the power side, the low voltage side should still provide a steady voltage to the appliance.
Devices that use unregulated analog power supplies are different, however. Doubling the voltage to the power supply will result in double the power to the appliance. If the power supply does not blow first (or fuses), then the appliance itself might be damaged. I suggest just leaving the original cord/lead in place. That way when someone sees it, they will not get confused about the power requirements of that device.