I've done a lot of research on this topic as it pertains to Italian nationalists who emigrated to the US in this period, rather than Germans, but think I can offer some useful perspective.

The failed German revolution was part of a wave of successful and unsuccessful revolutions in Europe, and many of those revolutionaries sought to gain support from Americans by appealing to the language of American revolution. Lajos Kossuth, for instance, went on a tour around the U.S. after the revolution he led in Hungary failed, in attempt to garner American support. He didn't get the military or financial support he wanted, but was well received by American audiences, who generally loved the idea that their country's existence inspired European action and that their country's founding was a historical trigger point and that all other nations would eventually sculpt themselves in America's image.

During the Civil War, both the Union and Confederacy sought to persuade European powers over to their side, often by claiming they were the true representatives of American democracy. This was a particularly important, but ultimately failed endeavor on the part of the Confederacy, because while declaring independence is often conceived of as the act of one country (or one group of rebels), true nationhood is reliant on formal recognition from other nations, who agree to lend to that new nation, sign treaties with it, etc. In fact, the desire to keep European countries from recognizing the Confederacy significantly factored into Lincoln's decision to deliver the emancipation proclamation, because it kept countries like Britain, who were happy to see the US split, from being able to justify Confederate recognition to their people.

Crucial to this PR campaign were (typically elite, wealthy) immigrants from 48, who served as important if often unofficial diplomats. For example, a very well connected Italian turned New Yorker named Vincenzo Botta made the case for the Union with the Italian king, visiting diplomats, and in a series of articles published in Turin. Less elite European-born Americans also helped carry this message back to Europe. Out of New York, the Garibaldi Guard was a military regiment named after a popular Italian Risorgimento figure and comprised of immigrants from countries like Italy, Germany, and Hungary.

In sum, European revolutionaries from '48 and Americans in the Civil War often tried to appeal to each other with the language of democracy and revolution in attempt to achieve their own political aims, and the immigrants who represented both groups often served as important facilitators and symbols of this exchange.

Thanks to /u/Goat_Island_Goat for their answer. I study German immigration to the United States specifically, so I'll chime in here too

This is a big question, so I'm undoubtedly going to be papering over and omitting things. The short answer to "what impacts did they have on Civil War Era American politics" is that they probably didn't have much of an impact, but it's really hard to tell, because as you noted, there were hundreds of thousands of German-speaking immigrants who came to the United States in the period before the Civil War. However, Bruce Levine estimates in The Spirit of 1848: German Immigrants, Labor Conflict, and the Coming of the Civil War that only around 2,000 of the immigrants who arrived in the wake of the 1848 Revolutions can really be considered 48ers in the standard sense of the term - namely, politically active, generally radical types. Of course, the general political instability caused by the revolutions spurred more who may not otherwise have felt the need to leave, but it's also hard to separate that from the potato blight that was hitting northwest Germany around the same time.

So what effect did the 48ers actually have when they came to the United States? As you would imagine, the majority of 48ers settled in predominantly German-speaking communities, where they encountered mixed receptions. Mischa Honeck describes several prominent 48ers and their political activities in We Are the Revolutionists: German-Speaking Immigrants and American Abolitionists after 1848, and the standard story for most bona fide 48ers is a life of disillusionment and a laundry list of failed newspapers and periodicals. The major obstacle to political integration within the German-speaking public sphere was an ongoing conflict between the so-called "Grauen," German immigrants who had mainly arrived in the 1830s, and the "Grünen," or those who arrived in the wake of 1848. By and large, the newcomers were seen as rocking the boat and out of step with American political conditions; one German-American writing at the turn of the century wrote that this tension only resolved after several years in the United States, after which 48ers "became most valuable citizens of the republic, as soon as their honest, but, under the circumstances, purposeless enthusiasm had changed into the sober endeavor to secure an existence by hard work and industry.” (Georg von Skal, History of German Immigration in the United States and Successful German Americans and Their Descendants, 29). This was the consensus among historians (mostly amateur historians) of German America in the nineteenth century; Der Deutsche Pionier, a historical periodical published in Cincinnati but circulated nationally in the 1870s-80s, counted among its contributors basically a "who's who" of German-American intellectuals, and the consensus was that most 48ers were essentially well-intentioned, but hare brained. They did credit them, however, with generally raising the level of political consciousness of German-Americans, especially as it pertained to German membership in the Republican Party, of which more below.

To the extent that there was a concrete result from 48ers' political activity, it was mainly due to the actions of specific individuals who were able to bridge the gap between German- and English-speaking communities. Carl Schurz is by far the most famous of these individuals, although Alison Clark Efford details several more in German Immigrants, Race, and Citizenship in the Civil War Era, as does Daniel Nagel in Von republikanischen Deutschen zu deutsch-amerikanischen Republikanern: Ein Beitrag zum Identitätswandel der deutschen Achtundvierziger in den Vereinigten Staaten, 1850-1861. If there is a single accomplishment you attribute to this group of 48ers, it was in creating an archetype of what Efford calls the "freedom-loving German." In essence, Schurz and others were able to tie the revolutionary ideals of 1848 to the free soil/free labor/antislavery views of the emerging Republican Party in the 1850s. This was not an easy task, specifically because of the prominence of former Know-Nothings in the GOP. That's why the antinativist clause in the 1860 Republican Presidential Platform was known as the "German Plank" - politicians like Schurz, and especially Friedrich Kapp, New York City's most prominent 48er, were able to convince large numbers of German-Americans to vote Republican only by emphasizing that the Republicans were sufficiently distanced from the Know-Nothings that they could unite around their other policies.

This gets to the question of how effective the 48ers actually were, though. For a long time, something called the "ethnocultural thesis" held that after being reluctant to vote Republican in 1856, German-Americans flocked to support Lincoln in 1860, and this move, specifically, proved decisive in Lincoln's election. On the face of it, there's a lot to support this; Lincoln famously bought the Illinois Staats-Anzeiger, Carl Schurz was employed as a stump speaker, and organizations such as Turner Societies (technically gymnastic societies, but more important as radical political groups) turned out publicly to support him. The problem is, the numbers just don't support it. At best, German-Americans in the North were split about 50/50, with significant regional variations. It's also worth noting the dulled impact of individual personalities. Carl Schurz was already famous by the mid-1850s and would be by far the most prominent German-American of the 19th century, but in his home state of Wisconsin, German-Americans went for Douglass. In New York City, where Friedrich Kapp spared no effort on Lincoln's behalf, fully two thirds of the German wards went against him. (See Walter Kamphoefner, “German-Americans and Civil War Politics: A Reconsideration of the Ethnocultural Thesis”). So while there was definitely a shift between 1856 and 1860, it's hard to figure out how much of a shift, and how much credit you can give to 48ers' political activism.

That, I suppose, gives a very problematic and highly inaccurate possible answer to your question as to what percentage of German-Americans would have been sympathetic to 48ers. In the end, there's really no way of knowing.

Where we do know 48ers had an outsized impact within the German-American community was in mobilization upon the outbreak of the Civil War. That's a long story in itself, which I will write more on if you're interested, but I'll refrain for now because ideologically speaking it's not substantially different from the ideology of the "freedom-loving German."

As far as settlement patterns are concerned, I'll give an oversimplified run down of the two broad categories of 48ers. The smaller group were so-called "Latin farmers." These folks were intellectuals who generally had little to no actual agricultural experience, and the ones who moved to the Texas hill country (Friedrich Kapp's uncle, Ernst Kapp, settled here and is representative of that colonizing movement) and parts of Missouri and Wisconsin. By and large, they didn't succeed in these farming endeavors. The rest settled in urban areas, where their talents as journalists, lawyers, and such were more useful. New York was an obvious choice because, even though we don't think of it as a largely ethnic German city, it contained the world's third largest urban German-speaking population in 1860. Other cities like Milwaukee and Cincinnati were popular simply because of how large their German-American populations were; the former was more than half German, and as one German put it "here is the most splendid company, beautiful women, splendid walks and boat rides, the most ambrosial air, and - last but not least - splendid lager beer. What else could a German want to feel at home?" (Rudolf Cronau, Von Wunderland zu Wunderland: Landschafts- und Lebensbilder aus den Staaten und Territorien der Union, 5). Casual misogyny aside, you can see the appeal of that environment, especially for someone who did not leave Germany of their own free will.

In terms of further reading, I'd recommend any of the stuff I quoted above, particularly Efford's work. If you have any specific topics you'd like to look into more I'd be happy to come up with more titles. Hope this answers your question, even if the answers are a bit dissatisfying.

Edit - typos