This answer needs to be proceeded by three disclaimers. The first is that, as you noted in your question, there were a lot of German speakers in the United States at the time of German unification, which makes it impossible to speak of any singular German-American reaction. Almost all histories of German-Americans acknowledge this fact, although some do a better job of reminding their reader that they are not speaking for all German-Americans as the work goes on than others. The second disclaimer follows from the first, which is to emphasize that I personally will be focusing on urban German-American communities. The third disclaimer is that I'm currently on the road and my references to secondary literature are therefore unfortunately slightly less specific than I would like.

The consolidation of German-American identity before German unification is quite complicated, and extremely uneven. German speakers have been in North America since before the American Revolution, and if the Anglo-Americans Peter Silver quotes in Our Savage Neighbors (2008) are any indication, English speakers at least collapsed "Germans" from all regions into a singular group (although as Celia Applegate notes in A Nation of Provincials, they also applied the term Pfälzer to all German immigrants, regardless if they were from Pfalz, which suggests that the external collapsing of group identities into a singular identity was not concerned with a "German" identity).

It is unclear to me if that same phenomenon extended to colonial era Germans themselves (and if anyone could chime in on that point I would greatly appreciate it), although many German-Americans within living memory did not think it did. In 1847, a visitor named Franz Löher wrote Geschichte und Zustände der Deutschen in Amerika (History and Present Conditions of Germans in America) in an effort to record the history of German-speakers in North America. Löher viewed the first wave of German immigrants as primarily consisting of religious refugees, with a handful of "adventurers" scattered in. The great change came in the wave that surged in the 1830s, which according to Löher enjoyed a "German" consciousness that had been lacking in earlier generations (pg 230). The Wars of Liberation against Napoleon reportedly played a large role, and although Löher did not use the term "nationalism," he does argue that the new generation considered themselves culturally German. Thus when Löher looked around Cincinnati in 1846-47, he saw "true German men, German by education and German in sensibility” (p 330).

Other observations both confirm and complicate this. Residents of antebellum New York's largest German enclave called their neighborhood Kleindeutschland, or Little Germany. As Stanley Nadel notes in Little Germany: Ethnicity, Religion, and Class in New York City, 1845-80, however, German-speaking New Yorkers were more often than not inclined to reproduce the religious and geographic divisions of Europe, so Badeners lived with Badeners, Brandenburgers with Brandenburgers, and so on (pgs. 37-42). A similar dynamic was at work in Cincinnati. At the moment I'm working on German-American militias in antebellum Cincinnati. Among those were units named the Deutscher Bürgergarde and the Deutsche-Jäger. Obviously it would be impossible to call themselves "German" if there were not a preexisting, if contested, notion of what is "German" or "Germanness." At the same time, Max Bergheim records in Cincinnati in Wort und Bild (Cincinnati in Word and Image) that an 1844 May Festival featured and exhortation from a local priest to put aside their peculiar differences for the sake of German unity, which indicates both a desire for that unity and difficulty in obtaining it (pgs 85-86). Wolfgang Grams also has an interesting case study which demonstrates that religious institutions were similarly fractured. See “The North German Lutheran Church in Cincinnati: an ‘Osnabrück’ Congregation,” in Eberhard Reichmann et. al, eds., Emigration and Settlement Patterns of German Communities in North America (Indianapolis: Max Kade German-American Center, 1995): 79-89. To complicate matters even further, you have the 48ers, who arrived in the thousands after the failed revolutions of 1848. They are a motley assortment with a wide range of political views and are plentiful material for a subject all their own, but the most common similarity is that as revolutionaries they fought for a united Germany and maintained a hope for one after they were exiled to the United States. Despite constantly butting heads when it came to specific policies, 48ers took an outsized role in public life, and particularly in journalism, which meant that they had an influence (albeit one that is difficult to quantify) both in getting German-Americans into American politics and in raising more interest in a unified Germany. For more on them, see Bruce Levine, The Spirit of 1848 (1992); Mischa Honeck, We Are the Revolutionists (2011); and Daniel Nagel, Von republikanischen Deutschen zu deutsch-amerikanischen Republikanern (2012), among many, many others.

So what happened when Germany unified in 1871? One of the best accounts of this event is a chapter in Alison Clark Efford's German Immigrants, Race, and Citizenship in the Civil War Era (2013). Efford's overall argument is fascinating and extremely well argued, but goes far beyond the scope of your question here, so I will merely refer to her chapter "Wendepunkt: The Franco-Prussian War, 1870-1871." Efford shows that the Franco-Prussian War was the subject of intense interest in the German-American press, and, at least in urban communities, the object of various rallies and fundraising efforts to support Prussia's war effort. There were obviously dissenters, but these people were generally silenced quickly by either social pressure or by the overwhelming successes of the Prussian military.
It is essential to note, however, that by and large German-speakers in the United States did not identify as "German," but as either fully-assimilated "Americans" or as "German-Americans." The last category is the subject of intense study, particularly in the period between the Franco-Prussian War and World War I. People who valued a "German-American" identity held that they were neither German nor American, but a unique combination of the best aspects of both. I would be happy to go into more detail on this portion if you would like me to, although I'm not sure at this point if that's part of your question or if you were more interested in 1871 itself, so I'll stop here for now.

Hope this is helpful!