How much was the engine moved in the B737 Max w.r.t. older versions?
The Boeing 737-800 and 737 MAX 8 both have the exact same fuselage length of $ 39.47 \, \mathrm{m} $, so I chose these two models for comparison.

Boeing publishes CAD drawings for all aircraft models. I used the two CAD files for the 800 and MAX 8 to compare the engine size and location:




Since I do not know the units used in the CAD files, I used the known total fuselage length to scale the numbers:


Distance from the nose to the front of the engine nacelle:


$$ \text{737-800:} \; 13.544 \, \mathrm{m} \; , \; \; \text{737 MAX 8:} \; 13.180 \, \mathrm{m} $$


Length of the engine nacelle including exhaust:


$$ \text{737-800:} \; 5.075 \, \mathrm{m} \; , \; \; \text{737 MAX 8:} \; 5.345 \, \mathrm{m} $$


Diameter of the engine nacelle:


$$ \text{737-800:} \; 1.978 \, \mathrm{m} \; , \; \; \text{737 MAX 8:} \; 2.407 \, \mathrm{m} $$


Distance from the top of the engine nacelle to the bottom of the fuselage:


$$ \text{737-800:} \; 1.359 \, \mathrm{m} \; , \; \; \text{737 MAX 8:} \; 1.554 \, \mathrm{m} $$

Based on these measurements, the new engine nacelles are $ 270 \, \mathrm{mm} $ longer and have a $ 429 \, \mathrm{mm} $ larger diameter. They start $ 364 \, \mathrm{mm} $ further forwards and the top is $ 195 \, \mathrm{mm} $ higher.

Therefore, the center (geometric, not necessarily center of gravity) of the engines has moved $ 229 \, \mathrm{mm} $ forwards and $ 19 \, \mathrm{mm} $ downwards. While the geometric center may have moved downwards, the center of gravity probably moved upwards because the old nacelle was not symmetric (flattened at the bottom).