Although the current German Basic Law has enshrined the right to conscientious objection, the Third Reich's legal system had no such provisions. Instead, the state enfolded conscientious objectors into the category of Wehrkraftzersetzer (underminers of morale), a group that also included deserters, chronic shirkers, and other malcontents. Defendants accused of being Wehrkraftzersetzer were not allowed to state their moral or political reasoning for their actions during their trials. This had made it somewhat difficult to identify specific instances of conscientious objection within the Third Reich (with the notable exception Jehovah's Witnesses were a special case in that they are much easier to identify and were the largest social bloc of Wehrkraftzersetzer). Article 48 of the military penal code had affirmed that German's duty to obey superseded any needs of individual conscience, religious or otherwise.

The punishments for Wehrkraftzersetzer in both military and civilian courts was quite severe and often became more brutal as the fortunes of war turned against Germany. But the harsh attitude of the Third Reich's legal system towards objectors was often quite severe from the beginning. Punishments for this offense could range from military discipline, a sentence into a concentration camp, or execution. In the latter case, military tribunals gave out 112 death sentences for Wehrkraftzersetzer during the first year of the war alone, and passed out approximately 23000 death sentences for desertion by 1945, of which they carried out some 15000.

For its part, much of the Wehrmacht's military leadership approved of the harsh penalties for Wehrkraftzersetzer. Although the Dolchstoßlegende (stab in the back legend) has been thoroughly disproved today, it held a great degree of currency among the military leadership of the Third Reich. Even for officers with little love for either Hitler or the NSDAP took it as a matter of pride that their units did not suffer an ignominious collapse as in 1918. Therefore, a number of Wehrkraftzersetzer were segregated from allegedly "healthy" units and placed either in special punishment units (Strafbataillone) for reform or otherwise punished. The attitude of Wehrmacht jurists was that individuals brought up on Wehrkraftzersetzer charges were somehow mentally deficient or otherwise lost their nerves and such defeatism could spread into their comrades.

So while your grandmother's story might be somewhat embellished (as family stories are wont to do), there is a good degree of veracity to it and the core narrative is not entirely outside the realm of possibility. Although some authorities were slightly more amenable to conscientious objectors in both the prewar and early war periods, sometimes offering individuals opportunities to serve in non-combatant units, this attitude was an increasing rarity as the war dragged on. The punitive nature of the Third Reich's punishment of Wehrkraftzersetzer is illustrative of what the German emigre Ernst Fraenkel called the "prerogative state" in which legal precepts and punishments are adapted to suit the needs of the state and not the individual rights. This allowed for a legal system that enabled arbitrary and violent punishments that grew more extreme as the state's existence was on the line.

Sources

Faulkner Rossi, Lauren. Wehrmacht Priests: Catholicism and the Nazi War of Annihilation. Cambridge, Mass: Harvard University Press, 2015.

Stargardt, Nicholas. The German War: A Nation Under Arms, 1939-1945 : Citizens and Soldiers. New York: Basic Books, 2015.

Wette, Wolfram. The Wehrmacht history, myth, reality. Cambridge, Mass: Harvard University Press, 2007.