I'm going to attempt to answer this. That said, I'd like to preface my response with the suggestion that nationalism outside of Germany (specifically in Central Europe) played just as important a role as German nationalism.

Germany as a unified state is a new concept, only going back 150 years. During the days of the Holy Roman Empire, the area we think of as Germany consisted of a patchwork of kingdoms, duchies, principalities, and free cities numbering in the hundreds. These territories were consolidated over time, and when Napoleon dissolved the HRE in 1806, he continued to redefine the borders of states within Germany. After Napoleon's defeat, the Congress of Vienna (1815) clearly defined 39 German states existing in a loose confederation.

Of course, despite its division into many different states Germany was still largely ethno-culturally homogeneous. If you were to speak to a citizen of any of these 39 states, they would in all likelihood speak German as their mother tongue and identify themselves as German. The main division between the German states had to do with religion: Catholic or Protestant.

If you go south a little bit to Austria, things begin to change. Although Austria was ruled by a German elite, it was a supranational empire with no fewer than 11 subject nationalities. Unlike other states, who tend to expand through colonization or military action (and thus have some degree of ethno-cultural consistency), the Austrian Empire was the product of dynastic marriages that had accumulated large swathes of territory over the centuries. The only thing that these disparate peoples had in common was an allegiance to the ruling house of Habsburg.

After 1848 (the year in which Europe was rocked by a series of attempted revolutions inspired by the French one), the Austrian Emperor Franz Josef tried to walk a fine line between preserving the power of the monarchy and keeping his subject nationalities happy to prevent more revolutions. The Compromise of 1867 (the Ausgleich) that elevated Hungary to equal status with Austria and created the Dual Monarchy is one example; another was the new rule requiring bureaucrats in Bohemia to speak both Czech and German, which provided native Czechs with an advantage when entering the civil service.

Actions like these angered Germans in the Empire, who felt threatened by the concessions granted to other nationalities. Coupled with German unification to the north, this gave rise to a Pan-German movement, a nationalist, right-winged, anti-semitic movement that wanted to shed the non-German parts of the Empire and join with their Teutonic brothers to the north. The Pan-German movement would later be a great inspiration to Adolf Hitler, who saw its leader, Georg Schonerer, as a hero. This aspiration was eventually realized during the Anschluss of 1938, in which the rump state of Austria was willingly absorbed into the Third Reich.

But going back a little bit. Nationalism also contributed to the outbreak of WW1, both in Germany and the Balkans. Prussia seized Alsace-Lorraine from the French during the Franco-Prussian war leading up to the German unification. Alsace-Lorraine had changed hands a number of times over the years and had both a French and German population, so both France and Prussia saw themselves as having a legitimate claim to it. The Prussian takeover of Alsace-Lorraine earned them France's lasting enmity and gave France a revaunchist motivation for war with Imperial Germany.

Im the Balkans, Austria-Hugary's annexation of Bosnia-Herzegovina quashed Serbian aspirations for a greater Serbian state incorporating those territories. Slavs within Bosnia saw the Austrians as foreign oppressors, and the Black Hand (which assassinated Archduke Franz Ferdinand in Sarajevo) was a Serbian-linked Slavic nationalist group.

Italian nationalism and an irredentist desire to seize Austria-Hungary's last remaining Italian possessions (Trieste and the Tyrol) also provided an incentive for them to remain neutral at the outbreak of the war and then join the Entente against their former allies.

Fast forward to the war itself. Every nation carries in it the seeds of its own destruction. For Austria-Hungary, its multiculturalism was one of those seeds, though it would take the external strains of WW1 to see them grow. During the war, Entente propaganda began to characterize Austria-Hungary as "the prison of nations", and nationalists such as Tomas Masaryk advocated for the dissolution of the monarchy. In his 14 Points, Woodrow Wilson stipulated that, "The people of Austria-Hungary, whose place among the nations we wish to see safeguarded and assured, should be accorded the freest opportunity to autonomous development," effectively sanctioning the partition of the Empire into national states.

Austria-Hungary dissolved into these new nation states after WW1. This proved problematic: although the various parts of the Empire had been dominated by certain nationalities, cross-pollination had occurred over the centuries. Consequently, there were large enclaves of other nationalities within these new Central European states. This would result in terrible clashes along ethnic lines that have persisted up until this day. These divisions within Central Europe made it susceptible to the Third Reich.

Another factor was the anti-communist hysteria after WW1. Many in Europe feared the creation of new communist states out of the wreckage of Austria-Hungary. In Vienna, armed members of the Red Guard (an extreme branch of the communist party) tried to storm Parliament and ripped the white stripe out of Austrian flags. In Germany, the Sparticist Uprising pitted the interim government against a communist force to decide the future of the state. Failed communist putsches like these made for fertile ground for facist leanings within the new states of Europe.

To recap, nationalism in Germany and in Central Europe was contributed to both world wars for the following reason:

• German unification led to direct conflict with its neighbour to the west, France, whom it would oppose in both world wars
• Central Europe was not homogeneous in terms of its ethno-cultural makeup, meaning that the goals of one nationality in the region would often clash with another's
• these divisions in Central Europe meant that Austria-Hungary had to make concessions to its subject nationalities, fuelling German nationalism within Austria-Hungary and providing a precursor to National Socialism
• the collapse of Imperial Russia during WW1 and the rise of the Soviets gave rise to a fear of communism that helped make Europe amenable to facist tendencies, further enabling the rise of the Nazis (though it's worth noting that at their start, they were a fairly minor party in the Weimar Republic)

I may go back and try to refine this answer a little bit, but I hope this goes part of the way to answering your question.

Source: "The Fall of the House of Habsburg", Edward Crankshaw. "Last Days of Imperial Vienna", Robert Pick.