Above: The Wannsee Protocol's estimate of Jewish populations in Europe. Subtracting the USSR, the UK, Ireland, Latvia, Lithuania, Estonia and Spain gives a total of 5.6 million.
I agree with the above: ghettoization was well underway by June 1941, and people were already dying in the tens of thousands, but the Einsatzgruppen and mass organized deliberate murder started only after Barbarossa. Given
Hitler's January 1939 "prophecy" -- that a world war would lead to the annihilation of the Jewish 'race' in Europe (
die Vernichtung der Judischen Rasse in Europa) -- I think the Final Solution always was, at some level, one of Hitler's chief war aims. It will happen without Barbarossa, although it would take on a different form. Giving a good answer to the WI requires, IMO, a date for the armistice, so I'll presume the end of the war occurs between 22 June and 7 December of 1941. It looks something like this: the UK retains all territory, including the Channel Islands. France, less Alsace-Lorraine, is nominally independent with a reduced military under a provisional government headed up by Pétain and Laval, with (free) elections scheduled for 1946, 5 years on. All of Poland that was once part of Germany is restored to the Altreich, Czechia remains annexed, but the Generalgouvernement will remain occupied territory until 1951. All of Central and Eastern Europe is under either Nazi or Soviet control, and relations are such that no satellite state dares to interfere with the Final Solution. The Wehrmacht would withdraw from occupied territories in 6-12 months from the date of armistice -- unless, of course, the Hitler-aligned leaders of a country requested the Wehrmacht to stay for "security reasons." The Waffen-SS would remain, making up virtually the entire military in the EBGs (ehemals besetzen Gebieten, formerly occupied territories).
I suppose that whole paragraph is OT, but I do think it's necessary to describe what the armistice would be like in order to comment on how the Final Solution would unfold.
Now, OTL Barbarossa was the event that "radicalized" the Nazi regime to mass deliberate murder rather than death by overwork, malnutrition and disease. Also IOTL, when the war began to go badly the Nazis stepped up the pace, and even when it was almost over they organized death marches. ITTL, the trigger would be different, but it would happen. IMO, the approaching end of the war is what causes these Nazis to begin "frenzied killing" of the sort OTL saw with the Einsatzgruppen and the Hungarian Jews. So it is the same trigger -- the end of the war -- but ITTL the Nazis are the victors.
In Poland, the Nazis could murder the Jewish population at their own pace, but Auschwitz and the Operation Reinhard camps would still be built. (Most Nazi victory TLs, for whatever reason, butterfly away the assassination of Heydrich, so perhaps Operation Gustloff.) The Nazis would have all the time they needed to kill the Polish Jews, so it's even possible they would operate only a single death camp, probably Auschwitz-Birkenau. If the OP intended the Germans to be Notzis, which I don't think was the intent, then mass sterilization and death by ghetto would be the way the Final Solution was conducted.
By 1944, as IOTL, the Jewish population of Poland would be some 10% of the 1939 level. By 1946, without fanfare, the Nazi leadership would be able to say Poland was
Judenfrei. Only a handful (thousands, if not hundreds) of
Partisaner and
Prominenten would remain alive and, as even the elite would be sterilized, the Jewish birth rate would reach zero. So, in a way, the Holocaust in Poland would be more complete than it was OTL, although the Soviet-occupied territory would not be affected, unless of course Stalin decides to pursue a Final Solution of his own, which is not entirely impossible.
The frenzied killing I mentioned would be seen in France and the Low Countries, primarily. Germany, having agreed to withdraw from those countries within 6-12 months, would set up Einsatzgruppen just as we saw in the East. Their modus operandi would be a little different -- once the Einsatzgruppen arrived at a village or town, they would lead the people to a remote spot and do the killing there. In larger cities, and as a supplementary method in the rural areas, mass deportations would occur. The destination would be the ghettos of Poland, then to the death camps when those were ready. Depending entirely on deportation is a possibility as well.
ITTL, would Sonderaktion 1005, the exhumation and burning of corpses and general destruction of evidence of the Holocaust, have happened? IOTL it began May 1942, a time in which Nazi victory was plausible (to them). I suspect ITTL it would be limited to France and the Low Countries.