The tube in the circuit looks to be for decorative purposes only and all the battery does is light the LED. I drew the circuit out and the only pins on the tube doing anything are the heater pins 4 and 5 passing the signal. So at best the heater element is being used as an inductor in the signal path. There is no current supplied to the heater so no space charge of electrons in the tube so how could the tube work? If you want to make a tube overdrive with low voltages they used to make tubes for car radios that work with a 12 volt supply. I have built an overdrive with one before and am now building a vibrato/tremolo and overdrive box with one now a 6GM8.
I can post a schematic of the posted circuit to give you an idea of what it is, some people would call it a signal canceling noise enhancing circuit.