I looked at the schematic for a while. Troubleshooting will be easier for you if you have a bit of background on how the two flashers tie into each other through the hazard switch. Forgive the long rambling about to follow.
Hazards are supposed to work anytime regardless of whether the key is even in the lock.
The purple wire going to the hazard flasher's "B" terminal is "hot" all the time, just like the horn.
The U/N (blue/brown) wire on the hazard flasher's "P" terminal is just for the indicator light on the dash.
The P/N (purple/brown) wire on the hazard flasher's "L" terminal goes to the hazard switch and then on to the lamps on the car's 4 corners.
Turn signals only work when the key is in the run position.
The G (green) wire on the turn flasher's "B" terminal is switched power from the fuse box.
The Lg/P (light green/purple) wire on the flasher's "P" terminal is for the dash indicator lamp, and
The Lg/N (light green/brown) wire on the flasher's "L" terminal is for the load (left and right signal lamps).
That Lg/N wire goes from the flasher to the turn signal switch.
When you select "left" or "right" the switch connects the lamps on one side of the car to the flasher completing a circuit and allowing the lights on one side of the car to flash.
The magic happens with the hazard switch. Study it on the wiring diagram.
When the hazard switch is "off", the left and right corner lights are separated from each other but connected to the turn signal flasher. When the turn signal is activated and the key is in the run position, power flows down the green wire, through the flasher, and to either the left or right side of the car (as selected). The current causes the flasher to heat and toggle the lamps on and off.
When the hazard switch is on, the lights on the left and right sides of the car are connected to each other AND to the hazard flasher. That completes a circuit through the hazard flasher so all the lamps flash on and off at the same time.
Again, study the switch and its connections carefully. Remember the purple wire should have power on it all the time. The dark green wire should only have power when the key is in the run position. When "off" the hazard switch separates the left and right side turn signal lights from each other. When "on" the hazard switch connects the left and right side turn signal lamps.
With all that said, there is more complicated wiring used on some cars. On some slightly later wiring diagrams the hazard switch completely breaks power flowing to the turn signal flasher when the hazard switch is "on". Since your car has had some conversions done, it could be that your switch needs to be different.