What is the dimension of the weak gauge field couplin constant NOT in natural units? What is the dimension of g and g', NOT in natural units, but in terms of mass, length, time, and permittivity?
 A: In natural units (first: there is an overwhelming reason and method in their madness!) these are both related to the electric charge e through the Weak mixing angle. See the triangle figure:
$$
g=e/\sin\theta_W, \qquad g'= e/\cos\theta_w, \\
 \sin^2 \theta _\text{W}=1-(m_\text{W}/m_\text{Z})^2=0.2223(21).
$$
All you need then is to convert the natural units e to the engineering units e you are seeking through the dimensionless fine structure constant, 
$$
α= {\mathbf{e}} ^2 /4π \epsilon_0 ħc =  0.0072973525664(17) ,
$$
or something... Insert your new/accurate CODATA values at your leisure.
In natural units, $\alpha = e^2/ 4 \pi$ since 1=ε0 = c = ħ = 1, so you redimensionalize the above g and g' by multiplying them with $\sqrt{\epsilon_0 ħc}$. 
At the core of your curiosity question, however is the 500lb gorilla question of why ? The weak interactions are microscopic, and you'll never monitor something about them with your ammeter, galvanometer, whatever. The microscopic world of HEP has precious few conceptual linkages with engineering. 
In an experiment, you will probably use the SM to compute lifetimes, in seconds, or cross sections in barns ($10^{−28} m^2$), etc, in which the dimensions of the couplings would be in the way---they are merely intermediate computational devices. (Your comment appears to trace to advantages of various systems in metrology, where HEP has little to offer.)
All you'd need do is throw in the suitable powers of ħ and c in redimensionalizing. 
For any intuition, stay natural and compare with your yardstick e=0.30282212....
