I can speak for Brazil.

The depression can be seen as the last nail in the First Brazillian Republic’s coffin. For 40 years by the time the depression hit, every single government failed to address the social problems and demands for basic worker’s rights. By that time, there had already been one attempted communist revolution, called the Prestes Column, and multiple revolts by the subversive lower class of the military, called “Tenentistas”, who, being young and coming from the lower classes, desired reform for the outdated way the First Republic operated.

The reason the First Brazillian Republic could afford to ignore all these revolts and popular demand for reform, was because it was a well-structured Oligarchy. The vote was restricted to only men who were able to read, which excluded most of the then-illiterate population. And from what elections there were, that were most certainly completely rigged, with classic fraud stumbles like there being more votes than electors. The ones behind the frauds were the agricultural elite, that controlled the country’s main export, coffee. By the 20th century, coffee consumption was reaching its apex worldwide, and the ridiculous money those big farmers made allowed them to control the political enviroment completely, with no one being able to stand up to them.

As the 20s rolled in, the price of coffee started to drop. That was bad news for the national economy, and for the big farmers that controlled the country. They started doing some pretty insane stuff to fix the crisis, like infamously burning the coffee they produced to keep the prices high, while the government paid them tax payer money as reparations for the lost merchandise. As the power of coffee dwindled, other elites sensed opportunity to capitalize on the widespread discontent of the populous and universal want for change to finally have a nation-wide opposition to the coffee planters in the upcoming elections. Of course, that was a first baby step, as they would need a lot more support to break the traditional power sources of the country.

Then, the depression finally hit. Of course, among the many things the people of the world stopped consuming in order to save money in these desperate times was coffee, bringing the depression to Brazil and collapsing the economy as well as the power core of the last 40 years of politics. Now, everybody was on board to take the candidate of the coffee lobbyists down with an unprecedented campaign in favor of opposition candidate Getúlio Vargas. It was a huge national event, everybody who was allowed to vote went out to vote, Vargas’ running mate was murdered and even the army was starting to consider abandoning the coffee producers... then the results came in, and Vargas was defeated.

To this day we don’t know how legit the result was for both sides, as both had signs of forgery, but following this there was a general uprising of indignation with the results. Army officials backed the revolters and began a march to the capital, declaring Vargas as the legitimate president. Vargas accepted the leadership of the revolution after it became clear the government was scared out of their minds and was losing ground to the revolutionaries, and joined the march to the capital, when the entire Brazillian army refused to start a civil war and defend the government, leading to a quick takeover by the revolutionaries. That would begin the “Vargas era”, that would cover 15 years total and see radical transformation to the country, but that’s outside the scope of your question.

So, basically the Great Depression was the final push needed to destabilize the already declining Republic and usher in a 15-year-long dictatorship in Brazil, that included reconciliation between Vargas and the coffee producers.

Sources:

Getúlio trilogy by Lira Neto

Oswaldo Aranha: A biography by Pedro Cortêa Lago

History of Brazillian liberalism by Antonio Paim

Eastern Europe

Traditional historiography of the Soviet economy held that the USSR was unaffected by the Great Depression and that the economy was thriving in spite of it. The second part is true, the first is not. The Soviet Union at the time of the stock market crash in the United States was in the middle of the highly successful First Five Year Plan which emphasized the production of capital goods (those that could be re-invested to create other goods, with steel being the prime example) and was able to overshoot its production targets. This and the following five year plan contributed greatly to the Soviet Union transitioning from an agrarian to an industrial economy before the outbreak of the Second World War. While industrial output grew at double digits, the overall growth rate of the Soviet economy was on the scale of the upper single digits when the agricultural sector is included.

The depression without a doubt affected the Soviet Union, however. According to earlier histories, the Soviet Union benefited from the Great Depression due to the ability to hire experts from the West, most prominently from Ford Motor Company, for cheap. These experts introduced to the USSR modern industrial management. However, the effect of this paled in comparison to the catastrophic drop in global food prices throughout the 1920s, one of the main causes of the Great Depression, and something that would continue during the 30s. Between 1926 and 1928, Soviet grain exports declined from 231,021,000 rubles to 51,512,000, owing both to greater diversion of food to feed the cities and falling grain prices . In the same period, imports of industrial machinery almost doubled to 247,300,000 rubles, driving the Soviet Union into a trade deficit by 1928. There is no way to objectively measure how much potential wealth was lost as a result, but it was certainly on the scale of hundreds of millions of rubles.

The Soviet economy, while successful on a numerical level, was at this point not offering its citizens a much better standard of living than before. Throughout the first two five year plans, industrial wages for Soviet workers actually dropped 17% on average. The emphasis for GOSPLAN, the Soviet central planning commission, was on increasing output, and the cheaper workers were, the better.

Hungary and Romania were two other economies dependent largely on exports of agricultural goods, with Romania also making revenue from exports of oil commodities. Falling grain prices led their governments to become largely insolvent, with the Hungarian government defaulting on its debt by 1931. Poland, meanwhile, was one of the hardest hit countries by the depression. Its economy attracted much foreign investment in the 1920s owing to economic reforms suggested by the renowned American economist Edwin Kemmerer. This made the contraction all the more severe, as many investors pulled out of the country in response to the market crash.

Asia

China escaped the Great Depression largely unscathed. The traditional explanation for this has been the maintenance of the Silver Standard whereas the rest of the world was on the Gold Standard. Recent work by Shiroyama Tomoko has discredited this hypothesis, proving that the value of silver increased relative to the value of gold in the early 1930s and hurt China's export competitiveness. The replacement of the silver standard in 1935 with fiat currency addressed this problem and allowed the Nanjing government to recapitalize the market.

Japan actually grew significantly during the depression. Whereas industrial output in the US and France declined more than 25% between 1929 and 1935, it actually rose by 42% in Japan during the same period. This has been attributed to the decisive response of finance minister Takahashi Korekiyo, whose policy of rapidly abandoning the Gold Standard, expanding the money supply, and expanding the state budget has been called "proto-Keynesianism". Aside from his financial stimulus, Japan also benefited from the occupation of Manchuria, whose industrial output by 1935 equalled that of Japan itself. In the puppet state of Manchukuo, companies connected to the Japanese army like the Manchurian Industrial Development Company (today known as Nissan) reaped immense profits using slave labor.

The depression in Japan and Manchukuo was extremely formative in the development of the post-war Asian "developmental state". In Japan, Takahashi had a lasting influence on the Finance Ministry and the Bank of Japan, and for the next five decades, both bodies would be controlled by "inflationists" who believed there was no end to printing money. The Manchukuo economic establishment, meanwhile, consisted mainly of "reform bureaucrats" with statist ideas who were gradually deported there by their more conservative colleagues in Tokyo. These bureaucrats re-aligned market incentives around their development goals and enforced oligopolies. The policies of the Japanese Finance Ministry and Manchurian central planners would be imitated by postwar Japan, South Korea, Taiwan, and eventually the PRC.

Like the Five Year Plans of the Soviet Union, the policies of Japan during the Great Depression have often been studied as a foil to unsuccessful recovery policies elsewhere. Some publications have suggested the Japanese bureaucrats synthesized a coherent Keynesian economic theory before Keynes, but this is wrong. In reality, most came from the Tokyo University Law School and had no theoretical understanding of economics whatsoever. This was the key to their success: they had no knowledge of the right thing to do, but also were not educated in the wrong thing to do. The economic mainstream at the time suggested that during a "bust", the market should fix itself and the government should focus on balancing its budget. These recommendations led to the tax and tariff hikes that worsened the crash elsewhere.

In India, overall economic output was relatively unaffected (with Angus Maddison estimating that Indian GDP actually grew from 1929-1934) but some industries suffered terribly. The worst effected were those depending on exports to Britain, which adopted protective trade policies. Five years after the 1929 market crash, Indian exports had fallen by more than half. India's paradoxical fate - an overall relatively healthy economy and severe decline in the export sector - had to do with its vast internal market. While industries depending on European consumption suffered, the standard of living for the great majority of Indians who were not fully integrated into the global supply chain did not change at all.

The situation was similar elsewhere in South Asia. In Burma, Indonesia, and the Philippines, essays by Peter Broomgard, Ian Brown, and Willem Wolters indicated that standards of living did not significantly decline despite the downturn in the global economy.

Africa

u/Commustar in this answer has already provided an excellent overview of the depression's impact in Africa. To summarize, in many parts of Africa, especially the Belgian Congo (in particular the mineral rich Katanga), Rhodesia, and South Africa, European colonizers had mobilized the local population to produce mineral and agricultural commodity products for export. These areas were heavily integrated into the global supply chain, and the collapse in commodity prices led to a wave of bankruptcies and layoffs in the primary sector. Colonial authorities increased extractive behavior in response to mounting debts, with policies ranging from agricultural purchase boards to forced labor.

While you wait for fresh answers from qualified experts, you may be interested in this previous response that specifically deals with the effects on African colonies, provided by /u/Commustar.