Since "Nationalsozialistische Deutsche Arbeiterpartei" is a mouthful even for a German, the National Socialist German Workers Party was shortened to the first two syllables of "National" in German, which is pronounced "Nah-tsee-oh-NAHL." It is that simple. The Nationalists, the National Socialists, whatever. Back then, just "Nah-Tsee" was enough for everyone to know what was meant. In Germany, although the French spelling of National is preserved when the word is fully written out, the German othrography (in German, "z" is the English "ts" ) prevailed with the abbreviation.
I wouldn't use "Putzi" since "puztig" is a term of endearment in German, sort of like "cute." Use "Putler" or some such. Or, as Trump probably used to call him:
I was discussing Putin's fading humanity with a friend who was a German news correspondent during the end of the Yeltsin years, and thus, Putin's ascendency. He said Putin was the scrappy poor kid who was long on street smarts and meanness. He has been "getting back" at the world for perceived wrongs ever since. He says that is how to look at what he has become, and we are right to fear him and wrong to ascribe any kind of reason or pity to him. He disdains compromise and pity. Compare him to Hitler? If he didn't despise Jews as much as any other ethnic group to which he doesn't belomg, he might just say, "Hitler, Schmitler, what do I care?"