There are multiple things to be discussed regarding LEDs and KVL
The term non-conducting is misleading. Even in reverse polarity a LED's internal resistance is not infinity but still has a (very) high resistance that can be measured or found in the datasheet as I-sub-R (reverse current). Replacing the LEDs and the DMM by their resistive values it is easy calculating their parallel (LED and DMM) and their serial resistances. It is a simple Voltage-divider circuit, but you should be aware that the voltage measured at each node is depending on the DMM connected parallel to it or not.

If you put DMMs parallel to each LED at the same time the sum of the voltages will add up to the input voltage. But still the measured voltages are different to those that exist without the DMMs.

E.g.: 100 M&Omega; parallel 10 M&Omega; = ~9 M&Omega;. 100 M&Omega; || 100 M&Omega; = 50 M&Omega;

If the DMMs internal resistance is not substantially higher than the internal resistance of the subject to be measured then the result is rather far away from reality.