The Low Countries (and Scandinavia) wouldn't exist in such a scenario, so their people would just get funneled into the Wehrmacht. Any nominally independent countries (shrunken France, Finland, Croatia, etc) keep their own militaries at a level just large enough for internal control and maybe fighting partisans in the East.
Finland is a special case here, because it will be significantly enlarged and will directly border Nazi-controlled territories in the former USSR. For a decade or two after the war is officially ended, at least, it will need larger than OTL peacetime forces just to enforce order in these areas. Given how overstretched the Germans themselves would be, the Nazi leadership demanding that Finland should send "police forces" to other parts of northern Russia would be likely. In comparison to the OTL, Finland would be seriously more militarized in the 50s and 60s, given that the Civil Guards organization will still exist as a national militia when it was disbanded on Soviet demands IOTL.
Generally speaking, as I think that the Reich proper can't keep a major number of its young men in uniform all the time just to hold its massive new empire together, I'd predict more demand for non-German allies'/satellites'/clients' soldiers in the New European Order. Whether these soldiers would be deployed (actually or nominally) in national uniforms or whether they would be put into the uniforms of an enlarged pan-European Waffen-SS in a "continental foreign legion" type of deal, or whether the setting would be something else, depends on what kind of a Nazi leadership follows Hitler. Under any circumstances, though, I believe that the Nazis would want the "lesser nationalities" doing the dirtywork of running their empire on the ground where possible so that the
true Aryans can enjoy the spoils of victory. In military terms, this would mean deploying foreign auxiliaries for guard and "police" duties, counterinsurgency, engineer units and other such less than glamorous jobs.