Measuring DC Currents
First: What is the resolution that you want to achieve?

Second: Before trying to reinvent the wheel, have a look at how high end measurement equipments are doing the job.

Usually the small res shunt is the used method. And when I said "small res", I mean ~100mΩ or less. Which is very small and should not be noticeable. The sensitivity of such measurements are very high if properly done. My Agilent DMM can measure current up to 100uA with 6.5 digit using 200Ω. If you don't need 6.5 digit the resistance can be reduced.
But 10uA in a 200Ω shunt has a voltage drop of only 0.2mV ! 

And you are right that, for measuring very small currents and big current, a high value resistor of 200Ω is not good. This is why a DMM has ranges. 

It could be of several kind:


Analog gain: You use a small shunt (~100mΩ) and you switch from one range to another by just changing the gain of the front end instrumentation differential amplifier at the shunt resistor. This has the benefit of allowing you to have several instrumentation differential amplifiers and when one saturates, you look at the result of the other ones. No relays, no data lost during range switching. The drawback is the noise. At some point with very low current and high gain you will basically measure the noise... Fluke portable DMM are using this method. But some has another input with another bigger shut resistor for small current measurement with less noise.
Shunt resistor range: You use a high-end relay to switch between different ranges of your measurement system. My Agilent 6.5 digit DMM is working that way. The advantage is the reduced noise problem, but the drawback is that it can't measure across the whole instrument range without switching relay and briefly opening the circuit.


Here are the shunt values used for my Agilent DMM:


200 Ω for 100 µA - 1 mA 
2 Ω for 10 mA - 100 mA
0.1 Ω for 1 A, 3 A


But first, you should define what is you required resolution. This would define the number of increments required to cover your full range (or sub range if using one of the proposed techniques) and thus the number of bit of your ADC.

If your require a resolution of 1uA with a full range of 1A, then you have to have an ADC with 1'000'000 steps: ~20bits !!! This is a lot! But if you have ranges of measurements, you could have higher res at low current and lower res at high current. And thus the required ADC would have much less bits.

To answer you question much in detail. You should replace your 0Ω resistors by a small shunt resistor that has the biggest value that does not disturb your system and then use a scope differential probe connected right on the pins of the resistor. Differential probes could be expensive but they include a differential amplifier that is placed close to the input probes. You could try to put a standard single ended probe at one pad of the resistor and another single ended probed at the other pad of your shunt, and then use the math of your scope to do the difference.This would be very noisy and may need some calibration to reduce offsets (especially if the scope was not recently calibrated).

This would give you an idea of the current consumption. As I already said, I don't know what is the resolution or accuracy requirements here.

Having the shunt on the low side is not advisable if you are measuring the power consumption of modules on a PCB. Good EMC/EMI/SI practices strongly suggest that you use continuous and strong GND planes. Segregating your GND plane and putting a single point connection between them using a resistor is a bad idea. because the return currents of the inter module communication would all be forced to flow through this resistor and would couple. This would be very bad in a signal integrity point of view.