Germany turned to Nazism after the oil crisis in Iran, with Heinrich Himmler becoming Chancellor by 1935, and Führer within months of Hindenburg's death (one year later than OTL). He spoke about "lebensraum" in the East, but with Iran and the USA going against him, that sentiment turned to Africa, with Germany annexing much of France's former colonial Empire as well as the fact that Africa was sparsely populated, anyway.
The Greater German Reich, many say, recovered well from the humiliating defeat of the Second World War (Hitler is an American---Version 1) and the nuclear destruction of Berlin. For starters its GDP, by 1948, rivaled that of the USA, Iran and Japan (which many say saved Göring's bacon). Many attribute its survival to the decision by Göring's predecessor Himmler not to attack the USSR and not to try to kill the Jews until its other enemies were taken care of first. However, that was not enough to save him. The Nazi party inner circle tried him (it is alleged he allowed corruption to spread in the Reich, and murdered his political opponents, but in reality it was a punishment for failing to win the war), found him guilty on all charges and sentenced to death by firing squad. The inner Circle appointed Göring to lead Germany (he was more competent, intelligent and not a drug addict ATL). He began the process known as the "Reich recovery plan", and appointed Albert Speer to be his Reichskanzler. Speer's first order of business was to try to rebuild Berlin. His plans made Berlin into the most modern city, with the Arch of Remembrance (the planned arch of victory of OTL) leading to the Volkshalle. Göring (with the approval of the Inner circle), commenced the space programme as part of his recovery plan.
However, since the Jews mainly went to America, the Germans then turned to the Native populations of their new African territories. What they did was declare them subjects of the Reich rather than citizens, which allowed them to be used as hard labor. To prevent them revolting, the Germans brought about "no more" than the South African system of Apartheid, used local native populations as foremen (which effectively said they were "Aryan") and set up the Afrikaan Arbeits Front, which was the same as the DAF OTL, except, the African laborers got less pay and less trips. Göring called his policy "the Carrot and the stick".
Germany's population exploded post WW2, with African territories providing a massive bounty of raw materials and
The Third Reich has a significant lead in the space race, with the Von Braun MkIII Stations (65% larger than the MkI and a lot more firepower) as well as the first man to Mars, beating the Japanese/American/Iranian effort by five years.
Germany recovered well from its defeat in 1943, with the GDP bouncing back above prewar levels within six years (mainly thanks to the mineral resources in Africa, which it ruthlessly exploited in the Congo and Mali)