To tag on to the post above, it is unlikely that all of France, for example, would have begun speaking German relatively shortly after after the fall of France in 1940. What is more likely - had Germany won the war - is that German would have become the administrative language if the conquered nations.

Since each country would ultimately answer to Germany, eventually anyone who wanted a career working with the government would learn German so that they would be able to communicate with the highest echelons of power. This is observable in colonial settings where locals would learn the colonial language for the same reason.

Now it is a harder question to say whether the average citizen would have spoken German after some time had passed. This would depend largely on the region. As Tim Snyder discussed in Bloodlands, the Nazi German aim for Eastern Europe was to either extermination or displacement of the local population and replacing them with ethnic Germans.

In Western Europe, Germany would have likely tried to exert cultural dominance as well though probably not to the degree of trying to wipe out the entire population. At any rate, it would be expected that everyone might be required to learn German in school or were heavily influenced to learn German. However, it is also notable that many well educated Western Europeans learned some German anyway. Also the Nazi ideology emphasized cultural and linguistic chauvinism; accordingly it is very likely that German language would have a “most favored language” status even in non-German nations of Western Europe.

So it is essentially unknowable whether Europeans would have all spoken German. In Eastern Europe, it is possible that there would have been no non-German population, or a much smaller one than pre-war. In Western Europe there would have been a strong German language bias in government and academia. If there really had been a 1,000 year Reich, perhaps there would have been a gradual shift to everyone speaking German but that is assuming everything worked out in Germany’s favor and every other Western European culture decided to give up on their own nationalism.

Your question is complicated by the fact the for most of recorded history the idea of a “national language,” or even a nation, did not exist. Invasion is one of the fundamental ways that languages have spread and evolved throughout history.

To answer your question, we’d have to limit ourselves squarely within the modern epoch of nationalism. In this period, language change became associated with ethnic cleansing, genocide and colonialism. Language was often used to determine national identity in cases of ethnic cleansing, as was the case in the Sudetenland after German occupation, and in the wars after the breakup of Yugoslavia.

Within colonialism, coerced language change was almost ubiquitous. Even in places like the Soviet Union, where local or indigenous societies were valorized, Russian was associated with literacy and pushed in literacy campaigns, thus leading to the extinction of some Samoyedic/East Siberian languages.