What Hades said and what I said are two similar but different things. I said that the Holocaust and the ongoing extermination of the Slavs would be public knowledge. He stated that it would be bragged about/glorified/celebrated. At least in reference to the genocide of the Slavic race.
I'm NOT saying that they would have festivals celebrating Auschwitz and Treblinka but that since the elimination of Jewry was a goal (not the only goal mind you) of the Nazis it's reasonable to say that the "annihilation" of Jewry would be mentioned even if just occasionally in propaganda and public speeches by Reich officials in the event of total victory in Europe. They did it IOTL often vaguely when the task wasn't completed so why would they stop when they've WON? They aren't scared of what the US and Britain will say about them in their newspapers. The Reich didn't care what others thought most of the time because they were genuine true believers in Nazism. Why should you hide what you're doing when you've already won and there's nothing any outside party can do to change that?
The extermination of Slavs would be common knowledge since millions of them would be transported to the Inner Reich and Western Europe as slave laborers annually, many of their jobs requiring German oversight or for them to work in the open (construction, roadwork, other manual/dirty jobs.) This all happened IOTL just on a smaller scale and not as deadly.
The German populace was used to seeing malnourished slave laborers (there were over 8 million slave laborers in 1944 or one fifth of the total Reich workforce) and Generalplan Ost clearly called for reduction of the Slavs through overwork, disease, and starvation so logically the German populace would have actual evidence that Slavs were being worked to death regularly. It would be about as normal and a fact of life as slaves on plantations in the pre Civil War American South were.
Would the extermination of Slavs be an explicit policy known to the public? Yes.
Would the German populace care much about what happens to "untermensch" and "vermin"? Hell no. Non Aryans existed to serve the Aryan race either through subjugation, enslavement, or death. This was explicit political doctrine going all the back to Mein Kampf in 1925.
Would the puppet governments of France, Belgium, Norway etc know that the Slavs were being slowly but surely wittled into oblivion through various means? Most likely considering that millions of laborers would be worked to death there as well (Atlantic Wall won't build itself).
Would the threat of their nation being turned into a charnel house/depopulated wasteland like Eastern Europe be used by the Reich to get their "allies" to follow orders? Probably. The governments of Western Europe will be pretty much over a barrel considering that the Wehrmacht/Waffen SS is the most powerful military force on the continent.
When you see a few hundred malnourished souls getting herded into a coal mine by overseers armed with clubs and whips (either civilians or SS) near your house on Monday and then when Sunday comes more than half of those workers don't come back at all and the survivors resemble skeletons, you're going to put 2 and 2 together. It's hard to deny mass murder when you can see the target group drop like flies right in front of your face.
A good question is why WOULDN'T the Reich admit their genocidal policies to their increasingly indoctrinated population? National Socialism, the Nazi Party, the SS, Hitler, and Hitler's inner circle were proven right because they won the war and achieved continental hegemony. Lebensraum and the Germanization of the East were explicit goals of Hitler and his henchmen and they knew that millions of Germans supported those policies as well. Hitler explicitly told his military high command on several occasions that the war against the Soviet Union was a "war of annihilation" between the "Master Race" and the Jewish led subhuman Slavic-Asiatic hordes.
In the event of a complete victory and the occupation of Europe from France to Norway to Greece to the Urals, they would double down on National Socialist ideology and doctrine because it was the entire POINT of the war. Not only was it the entire point of the war, the German populace would be convinced that Nazism was the reason why they won in the first place. This combined with the Hitler Youth and the League of German Girls plus other brainwashing/indoctrination programs means that as time goes on an ever increasing portion of the public (particularly those who can't remember when Hitler wasn't the Führer or young adults) will become Nazi zealots who practically worship Hitler and the Party and see the extermination and enslavement of Jews/Roma/handicapped/gays/Slavs as something that is necessary or even good for the prosperity of the Aryan race.