Measuring voltage drop across a diode in a switched circuit
How to measure the voltage drop across the diodes D102 and D103 in this senario?


Like you did: with an oscilloscoop.
For example by measuring the anode of the diode on one channel and the cathode on the other channel and use the math functionality (if exist) to calculate the difference, or use the cursor to measure the difference.
Another option would be to galvanic isolate the oscilloscoop and connect to probe across the diode.


  Why am I getting a PWM voltage waveform only the cathode of the diode and not at the anode?


Because the anode is connected to a voltage source which will regulate its voltage to maintain a constant 8V.


  Why is there a 0.3V voltage drop across the diode? Shouldn't it be typ. 0.7V?


Yes, it should.


Only when the ambient temperature would be 150&deg;C (which I doubt it is)  the drop would be 0.3V

UPDATE with simulation as discussed in comments
If the diodes are really 1SR154-400 and noty Schottky's, the only explanation for the \$V_F\$ of 300mV I can think of is that these diodes are hardly conducting. When assuming SBATT is a battery which voltage is about or slightly higher than 7.7V, then that battery provides the main current through R1. The contribution of the 8V is quite low, say about 2uA, resulting in a VF of only 300 mV. See simulation below. I used 2 shunt resistors of 1 m&Omega; to obain the current delivered by the battery and the current though the diodes. I made assumptions for the type of Q1 and Q2.