I don’t have a specific idea of what the problem is, but it sounds like some fundamentally bad connection somewhere which is creating a large resistance in the circuit. This will not necessarily be observable if you are just checking for voltages at various places because, in a circuit with no current, there will be no voltage drop.
Does your meter allow you to measure resistance (ohms)? I would check the resistance of various earths.
Also, do you know the condition of the battery? Doube-check the battery terminal connections, and check the battery is fine by connecting a high power item (headlamp, say) directly to the battery terminals.
What’s the charging circuit? (ie generator or alternator)? If generator, do you know that the control box, and the control box wiring is OK? If possible disconnect the wiring relating to the charging circuit, and just test everything with a fully charged battery, in case there is someting going wrong there. It will also eliminate the possibility that you’ve got a +ve earth loom on a -ve earth car or vice versa.
Then, test the simplest possible circuit, with as many bulbs removed or disconnected as possible. Also remove the hazard flasher if you suspect this. Say test the sidelights to begin with, then the headlamps. Avoid anything with a flasher unit to begin with because it makes things harder to measure. With the sidelights on, and then the headlamps on, measure the voltage drop from the battery +ve to the +ve terminal of the actual bulb, and similarly the voltage drop from the earth side of the lamp and the -ve of the battery. (I’m assuming you have -ve earth.) The voltage drops should be negligible (substantially less than 1v each), with virtually all the voltage drop across the lamp.
Hopefully this way you will either find a problem, or confirm that the basic wiring and distribution is OK. Now to the circuits that depend on the ignition (stop lamps, say, and the ignition itself).
Only put the bulbs back in to the circuits you’re about to test. Each time you put a bulb in, and test the circuit, re-test everything else with that new circuit turned on. This might identify a circuit that is shorting.
Report back if you find anything 
Paddy