I agree that R13 (and R12) are the volts measuring inputs forming a divider. You should check that the left side of R13 is connected to one volts terminal block and the bottom side of R12 to the other.
The large 47 Ohm R15 may be fusible (acting as a fuse), at least it acts to limit inrush to the power supply which by the luck of it is a buck converter to step down the voltage for the regulator.
Solutions
1 .In theory all, you need to do is disconnect the left side of R15 from the terminal block and connect it to a supply that is always at mains voltage. BUT (!!!!) this must be done in such a way that you do not create a short on the mains side. Say the terminal that used to feed R15 was Live and the other Neutral, then that Live terminal is the output of the VARIAC and the now disconnected R15 must instead be connected to the LIVE input of the VARIAC. There is a small possibility if that doing this will back-feed some voltage into the measuring side but I think it may work. If I were testing this I would run the input into the VARIAC through a current limiter using a 60 or 100W incandescent (!!!) light bulb wired in series. If the bulb lights up brightly, you have a problem that needs fixing. This is of course only for testing with no load connected to the VARIAC output. An example of a light bulb current limiter (super useful device to have) is here
https://youtu.be/EgRs2InB3_o2. To play it safe, an alternative is to disconnect R15 so the buck converter is now dead. Then build a second small power supply with a proper (isolated) transformer of say 12VAC, bridge rectifier and small smoothing cap of 220uF 25V and connect the output to the black and red leads in your picture to the regulator input. In either case, no earth is ever connected to the module.