I imagine if slavery were to continue into the 20th century there might be boycotts. Something akin to what happened with Sao Tome and Principe would happen. That year Britain's Anti-Slavery society investigated labour conditions on the island as labourers had been recruited in Portuguese West Africa to work on the cacao plantations. There investigations began around 1905 and the controversy ensued with some chocolatiers boycotting the island, and the Portuguese eventually made reforms to the system by 1909. This boycott was not uniform, as German, America, Dutch and other chocolate producers largely ignored the issue of slavery.
With cotton, the issue becomes a little trickier, as the former confederacy produced most of the world's supplies by 1913. Assuming the Confederacy remains intact, based on actual 1913 figures, it produces nearly two-thirds of the world's raw cotton, or 13,545,000 bales. India is next with 3,692,000 (17%), Egypt 1,496,000 (7%), Russia 1,030,000 (5%), China 620,000 (3%) and Brazil 420,000 (2%). Based on consumption, Britain consumed around 20% of the world's raw cotton, with the rest of Europe consuming another 20%, and the U.S. consuming around one-third of the world's raw cotton.
In terms of milled cotton, the U.S. share was higher, with 82% of all milled cotton in 1913 being produced in the U.S. another 17% in India, 7% in Egypt, 5% in China, and 3% in Russia. With textiles being such an important industrial export in many countries, this would give the Confederacy some leverage over control of the world market. Whether or not there are efforts to increase production elsewhere is up to anyone's guess. However, I imagine that for many European countries, they might simply choose to ignore the issue.
In 1860, U.S. Cotton exports were 3,535,373 bales, with 71% going to the United Kingdom, 16% to France, and 4% to the German States. By 1913, 9,124,591 bales were exported with 41% going to the UK, 26% to Germany, 12% to France and 15% going to the rest of Europe. This increasing share of non-British textile manufactures between 1860 and 1913 in a way would be beneficial to the Confederacy because they would not be so reliant on the British market for exports, and I can assume that is where most of the criticism would stem from (along with the U.S.).
In terms of textile manufacturing, by 1913 the UK was still dominant, it was home to 38% of the world's cotton spindles, the U.S. with 22%, Germany 8%, Germany 6% and Russia 6%. Interestingly enough, the ex-Confederacy was home to 11% of the world's spindles by 1913, an upward trend, perhaps due to lower labour costs. Of these, two-thirds was concentrated in the Carolinas and another 25% in Georgia and Alabama. Whether this would have developed as it did is anyone's guess, what is interesting is that African-Americans were largely excluded from mill jobs in the Carolinas.