It depends on what you mean by "uprising."
Were there large, coordinated, armed attempts to escape or destroy southern slavery during the Civil War, led and fought by enslaved people, as in the Haitian Revolution, Nat Turner's Revolt in 1831, or the Stono Rebellion in colonial South Carolina? No. But at the same time, black people, born free and enslaved, fighting in coordinated armed groups, played a decisive role in destroying slavery and ensuring a Union victory through their service in the Union army after the Emancipation Proclamation allowed for the enlistment of black soldiers. You probably knew that already though (happy to expand on this more if you were previously unaware of the large role that black Union soldiers played in the last years of the war).
But you've also touched on a long-running discussion in scholarship on American slavery: the question of "resistance." For a long time, historians were baffled at what they perceived to be the lack of rebelliousness of enslaved people in the United States, particularly after the American Revolution. Armed, organized rebellions occurred with relative regularity in places like Brazil, Saint-Domingue (which would later become the independent nation of Haiti after the most successful slave revolt in human history), and Jamaica. But in the United States, apart from Nat Turner's revolt in Virginia in 1831 and the German Coast Rebellion outside of New Orleans in 1811, there are few equivalent examples. This was a topic of great interests to historians of slavery in the United States.
Some, like Stanley Elkins, theorized that enslaved people were docile, uninterested in rebelling, and generally content with their situation and status. This argument is one that sounds repulsive to modern ears, but Elkins based his argument on a comparison of the experiences of enslaved people and survivors of the Nazi concentration camps. Psychological scholarship on survivors found that many of them experienced serious mental damage from the trauma they experienced, leaving them with cognitive and emotional handicaps that made it difficult for them to, say, form and maintain lasting human relationships (this scholarship dates from the 50s and 60s so is probably similarly out of date, but I haven't read widely enough to say). Elkins theorized that American slavery was so brutal, and that surviving it was so traumatic, that enslaved people were in effect forced into a state of arrested development. This was called the "Sambo thesis," named for the stereotypical black stock character.
Others, like Eugene Genovese, theorized that despite slavery's violence and cruelty, the ideology of the ruling slaveholding class was so hegemonic (i.e. that it so thoroughly dominated the mental universe of all southerners, white and black. this is a word he borrowed from Marxist revolutionary and scholar Antonio Gramscii) that enslaved people more or less accepted enslavement as legitimate, even as they fought back against what they saw as its abuses and demanded that their enslavers fulfill all of their own obligations.
In my view, and in the view of most contemporary slavery scholars, these two came to the wrong conclusions because they were asking the wrong questions and looking for the wrong things.
In the United States, a very effective regime of violence and terror discouraged large scale, armed, organized rebellions (though it could not stop them entirely). It was a regime that was more effective in the United States in part because the U.S. did not have the wildly disproportionate demographics of a place like Jamaica, where enslaved people were a decisive majority of the population. But that does not mean that enslaved people did not resist their enslavement. They did so constantly. These are only some of the ways that enslaved people fought back against their enslavers day in and day out:
-work slowdowns
-playing dumb
-dissembling
-lying
-faking sick
-destroying property or equipment through feigned incompetence
-destroying property or equipment through covert sabotage
-theft (though many enslaved people did not recognize this as theft in anything but the legalistic sense. Theft was a moral outrage if you stole from another member of the enslaved community. It was merely redistributing the products of their own labor if they robbed their enslaver. One enslaved man joked that since both he and a pig he stole and ate were owned by the same man, that this was literally just rearranging his property)
-temporary or permanent absention (temporary escapes were more common, particularly in the deeper south, but denying one's labor for days, weeks, or months could seriously damage the fortunes of one's enslaver, and sometimes serve as a way to wring concessions from them, even if they were empowered to torture you as a punishment if you returned)
-giving food, water, shelter, or information to people who had temporarily or permanently absented themselves
-abortion and infanticide (though the occurrences of this have probably been exaggerated by racist slaveholders who assumed that black women did not love their children in the same way white women did)
-suicide (more common among enslaved people who were not practicing christians)
-literally fighting back (though such an action could easily earn you a death sentence)
-arson
-murder
So, after that fairly lengthy preamble, we might wonder how the Civil War affected these patterns. It did so predictably: incidents of all of the following became more frequent and intense. About half a million people (out of four million enslaved in the south), made permanent escapes to Union lines. According to historian Stephanie Camp, the strategy of temporary escape that was seen so frequently before the war facilitated this by giving lots of enslaved workers a very detailed knowledge of the local geography and how to navigate it. Some of those who escaped to Union lines provided military intelligence to the Union officers they encountered.
Many enslaved workers, with their enslavers fled at the approach of Union soldiers, appropriated the personal property of their enslavers to themselves, and took up residence in the plantation's "big house." That is, if they didn't simply burn it to the ground instead.
And there was certainly violence directed against enslavers. The famous diarist Mary Chestnut -- a wealthy South Carolina slaveholder beloved for her thoughtful, sensitive, often biting account of the war, confederate leadership, and southern society -- recorded the death of her cousin Betsey in 1861. To Chestnut's horror, Betsey had been murdered in her sleep by some of the people she'd enslaved. Chestnut wrote:
I always felt that I had never injured any one black especially and therefore feared nothing from them - but now.
I suspect that Chestnut's self-assessment would not have been shared by the people she held as property, but that quote speaks to a new sense of anxiety and fear that slaveholders felt towards enslaved people as the war changed things. It is a profound irony that they could be so terrified after spending the last few decades insisting that the people they enslaved were happy to serve them and had no desire for freedom. Though there were plenty of slaveholders who went the way of Cousin Betsey, incidents of revenge violence were less frequent than in numerous other revolutionary conflicts. I think the reason for this is a bit understudied, but some possibilities include that decades of covert resistance were hard to break out of even in the presence of an apparent opportunity, that enslaved people felt that revenge was morally wrong and took their moral frameworks much more seriously than their enslavers did, or that violent revenge against the body of a hated enslaver was simply a low priority for any number of other reasons.