It still applies. For the raw AC, W for one half cycle is exactly the same as W for the next half cycle, so if you remove alternate half cycles (which is what the half-wave rectification does) you end up with W/2 average.
Considering a resistor as the load here, wattage is measured as the amount of heat, (work and/or horse power), produced or emitted from the load. With the half wave rectifier inline there is NO heating going on during the periods of no pulses; so there IS 1/2 less wattage than with either the full wave rectified signal that has the same amount of pulses as the full AC sine wave. The full AC sine wave, also has double the voltage in the circuit, because the voltage is measured top to bottom of any waveform, that is applied across the load. In this case (for the full 240v AC sine wave), 120v for the top + swinging side of the waveform, and 120v for the - swinging side of the waveform. (+)120v + (-)120v= 240v total. Where as, in the half wave or full wave rectified waveforms only swing + 120v from the "0" reference line. (Disregarding whether we are measuring the voltage at the peak, average, or RMS voltage level.)
So, for the half wave rectified signal or waveform. There is half the heat or wattage produced at the load, (the foot heating pads in this discussion), because there is half the voltage (120v) of the 240v input, and there is also only half of that lower 120v voltage heating going on at the load because, there are 50% less, or half of the "on" time periods or pulses to produce that heat. So, 1/2 of 1/2, equals 1/4 of the heat or wattage produced at the load.
As per ohm's law. P= (E x E) ÷ R, and/or P = (V x V ) ÷ R, where as "P" equals power in watts.
P= watts
E or V= voltage
R= load resistance
Also, these pictorial waveform drawings can portray either, the voltage or current swings in the circuit, because you need both voltage and current in a circuit to produce power. P= E x I.
ie. P (watts)= E (volts) x I (current/amps)