In the timeline of the New Order, a coalition of Chinese Nationalists and Communists managed to repel the Japanese from mainland Asia, severely weakening Japan's military and economic power. Korea was one of their last holdouts in the region. However, the Nazis invaded China through the conquered Soviet Union and forced the Japanese to relinquish their remaining territories in mainland Asia. Korea inevitably became one of these provinces that fell under the rule of the Greater Germania.
By 1961 Korea had been fully indoctrinated into Germany's Reich. Unlike most other regions of the world, the Nazis were not particularly brutal against the Korean people and did not persecute them for their race. However, they still subtely coerced the Korean people to largely abandon their historical identity such as following the teachings of Buddha. Despite this cultural imperialism, the Korean people viewed the Germans as benevolent liberators compared to the Japanese. Some Germans visited Korea for an adventurous vacation and made friends with the local people. Like elsewhere on the planet, the Germans rebuilt the capital at Seoul with impressive Uberconcrete superstructures.
Interestingly, it could be argued that Korea's fate in the world of Wolfenstein is somewhat similar to the real world's timeline. South Korea became newly industrialized, culturally changed, and linked into the larger world by foreign powers while North Korea retained its quiet isolated feel and built imposing buildings with fascist orientation. The only difference is that in our timeline Korea was divided after WWII.