As pointed out in the comments, the reason for this is the 2k2 resistor before the regulator. Because this is in series, any load will cause a voltage drop over it. A multimeter will use barely any current, hence you will see what appears to be a working circuit, however, even drawing just 10 mA will result in a voltage drop of 10*2.2 = 22V.
Remove this and your circuit should work fine.
Why it was added in the first place, I am unsure, my guesses would be perhaps for current measurement (because of the stated 1W), but had the value wrong, although even that would still have it placed incorrectly, it should be post regulator. The other thing is it could have been fitted wrong in the schematic and it was supposed to be in parallel with the smoothing capacitor to discharge it when powered off.
Either way, remove R1 and you should find your circuit works.