I would see this as a sequence of questions.
1. Could Nazi Germany win the war?
The odds were somewhat against them, but they could very well have won. There are many scenarios in which they could.
1A. A British separate peace
This could happen for many reasons. The British/French might renew "peace in our time" during the Phoney War. Alternatively a peace treaty after Dunkirk, or a successful invasion of Britain, or by attrition before Operation Barbarossa. If Britain was neutral, they might re-enter the war, but at a much later stage. If Britain was occupied or neutral, the US would be unlikely to enter the war, and wouldn't have a forward base if they did, and there would be a second Phoney War where Germany couldn't attack the US, nor the US Germany. With their backs free, not having to fight a two-front war, the odds in the East would be far better.
1B. A Soviet collapse
The campaign, delayed by Greece, could have started earlier in the year and the Blitzkrieg could have gained much more ground before being bogged down by Russian winter. If the objectives of Leningrad and Moscow were taken, it would be very hard for the Soviet forces to fight on and rebuild their production.
1C. Nuclear Nazis
If the Reich had had nuclear weapons in production, together with their rocket weapons, surrender would be likely. Alternatively nuclear annihilation of major cities and/or forces.
2. Would Nazi Germany have the genocidal drive and capability?
The Holocaust in OTL should show that they did. Only losing the war stopped that genocide. If they were not at risk of losing the war (1) the Holocaust would have been the pilot project. Experiences from that would be applied in Eastern Europe. Anti-Slavic hatred was not as universal in the Nazi party as antisemitism, but widespread enough, and it went to the top. The plans for Lebensraum was deeply founded.
However, a genocide of the majority population is more challenging than the minority, where they could use local hatred of minorities to their advantage. But 20th century history has shown it can be done, the Nazis wanted to do it, and there would be nobody to stop them. But it would be most easily done in Poland, Czechoslovakia, and Western Soviet Union. Particularly the Asian part of the Soviet Union would be hard to maintain control over, let alone organise a genocide, so the people living there would probably escape that fate. The other survivors would be slave labourers.
3. Would the Third Reich last long?
Long enough for this genocide, almost certainly. By the end of the war it had lasted 12 years, and it could easily last another 12. Would it survive the death of Hitler, or him becoming so erratic to not be capable of leading a Nazi Empire? Possibly. There could be an internal coup, keeping Hitler on as a figurehead (bit like Mao after The Great Leap forward. There would likely be a Cold War between the Nazi Empire, the US and other parties. The alliance with Japan would likely cool when not mutually beneficial. The story of the ongoing genocide would leak out. There would be outright rebellions as soon as they had any chance of success.
4. Would the Nazi Empire lose the Cold War, if so when?
They probably would, but it would most likely take decades. To pick a number, say 1965.
5. What would be the aftermath of the victory?
There were more vindictive voices after 1945, especially from France, as it had been after 1918. These would pale to the reaction after a horrendous genocide like this. The winning parties would agree that there should never again be a Germany. The country would be divided into multiple parts, something along the post-war model in OTL, but more permanent. The German industrial base would be destroyed, and Germany reduced to multiple agrarian colonies. Speaking German would likely be forbidden.
6. How long would it take Eastern and Western Europe to recover?
Stalin and the Soviet Union would no longer be a force to be considered. Communism might still be, but without the Soviet Union it would be less acute as a threat. Mao and the communists wouldn't win the civil war without Stalin's support, the nationalists would win the civil war. Without the Red Threat initiatives like the Marshall Plan would be less compelling. Colonies would have become independent during the Third Reich, or in its wake. Britain and France would not be able to maintain their old colonies. Reconstruction costs would be higher. The US would be weaker as well, but in much better position to recover, much like the 1950s, but Europe would lag (also slowing US growth).
Relative to OTL Eastern Europe would be non-totalitarian by 1965 rather than 1990, but after 25 years of Nazi tyranny, more destructive to society than the slower deterioration of Soviet socialism. On its own we should be about the same level by now, but in addition Europe as a whole would be worse off, and then there is the genocide. The aftermath of WWII in OTL gives an indication. E.g. in Czechoslovakia Sudeten Germans were evicted, a de facto ethnic cleansing. The areas in question have been lagging ever since, but some of the cities have been doing well. We could expect something similar in post-genocidal Eastern Europe.
