Kay nother won't hurt then
IMHO, Nazis (esp. those already searching for mystical power) would have the initial short term advantage due to the range & power of their first strike capability (depending upon size & nature of their initial forces), but would only retain that advantage by quick and early removal of key individuals (powerful spellcasters)...before they realize what that whistling sound in the sky or that flash of light from the treeline means. The Nazis would be going into each encounter in the new world with the mindset that they're encountering non-friendlies- not neccessarily the case with the fantasy world's natives.
The occult focus of the Nazi party came from a few members of the inner circle and was largely directed through Himmler who was himself very much into the occult aspects. It tended to be small specialized units within the SS under his orders who were sent out looking for occult objects. Such as the group sent up into the Himalayas and remote regions of Central Asia looking for the origins of the "Aryan" race.
So under this set of circumstances we can assume the Nazi Sorcerers within the effort to be a special unit of the SS. Probably detached and answering only to Himmler or one of his immediate staff members, on a direct order of the Fuhrer.
Once the rift was created things would get more interesting because unlike in our history now we'd have tangible results from the occult side. At this point anything could be on the other side, it's a complete unknown and thus the most extreme form of potential threat source given the occult nature of the rift. If this is happening later in the war it will be immediately viewed as a new line of retreat for Nazi forces. Even if happening earlier it will be viewed as a new and enticing source of lebenshraum.
A force will be mobilized to make the initial movement into the rift. This is nothing more solid than a WAG but I'd peg the initial force as probably equivalent to the invasion of Crete. There would be similarities in condition and goal, this would be a beachhead force. The other side would be unmapped and unknown with no infrastructure to support heavy or mechanized forces at first. Purely an infantry operation, likely carried out with either mountain troops or the Parachute Division they didn't reuse for that purpose after the losses at Crete. Both were highly trained light infantry trained to operate with limited material and fire support, while rapidly seizing or defending land through shock assault.
They would be treating any encounter as a meeting engagement. Can't blame them given they'd be stepping into a totally unknown situation. Even assuming they crossed in a remote area the "wildlife" of a typical D&D world is incredibly hostile. Regardless of the politics of the war and its aftermath German SS field force troops were highly competent on average. They would suffer casualties, but they would succeed in forming a beachhead.
Now for the real question: How would they deal with the natives?
This is speculation but I feel probably good speculation that they would likely kill Goblins, Orcs, and Bugbears on sight. Hobgoblins depending on how strong and large a population was encountered might be enticed into service as a client race or made slave-soldiers. This is largely a factor of how useful their ordered and regimented militarism would be. Might be they would just kill all the goblinoids on site but I think at least some they would feel usefull. Orcs are too chaotic for this. Halflings and Gnomes would depend on how they were encountered, the Nazi's didn't like Gypsies and if the halflings seemed too much like Gypsies might treat them similarly. On the other hand if they were a more hobbitish sort I can definitely see the Nazis leaving them to their farming and giving them a place as an underclass due to the psychological factor of their small size making them seem less of a threat and the need to establish sources of supply in the new land. Humans if encountered would go under martial law like any other German occupied territory. However their occultists would probably attempt to bring any local spellcasters onto their side. Those that didn't and weren't high level enough to use magic to escape would probably be executed as too great a threat to leave in the rear area. Elves would be enslaved, they simply do not fit into the Nazi social model and their habits and appearance would be far from the Aryan ideal being pushed. Dwarves would either end up at war or in an uneasy ceasefire. The Nazis would see a supply of badly needed resources, and if attacked the Dwarves would either die in their delves or retreat to the underdark. A trade in engineering and metallurgical techniques and such might develop.
The greatest question is how would they handle the truly monstrous intelligent races. Unintelligent monsters would be killed out of hand as the beasts they were to the Nazis. But some of the intelligent magical beasts and other would be an interesting fix. Depending on whether or not they could stop hostilities they might actually have a place for Giants. undead would probably be killed without hesitation, simply as a reaction issue. Outsiders in the prime material are variable but probably not well, angels are about the only exception and they don't tend to show up on the prime material that much. Dragons would be in trouble, every unit commander in the invading force would think no further than being known as a dragon slayer.
