Massive civil war. Hitler ensured there was no real process to succeed him and gave private armies roughly coequal in strength to a bunch of medium-functioning sociopathic mass murderers who hated each other’s guts and were in constant competition with each other by design in order to prevent coups.
Actually Goering was definitely (till about 1944) the second top dog in the Reich. While his standing
did suffer after the flop that was the Battle of Britain and the increasing Allied air war, he was never actually removed from his position as Hitler's successor until 1945 when Donitz was chosen in the moment when Hitler was going mad at the end. A Nazi German which (say in AANW fashion) reaches a state of detente with the West will probably have a Goering who remains powerful and Hitler's designated successor. His only real rival was Himmler, and both men trusted each other about as far as they could spit.
Honestly when Hitler dies, Goering assumes power, and if he feels he can get away with it he might purge the senior SS staff, but if he doesn't he keeps token SS guards but replaces them with his Luftwaffe ground formations. If Himmler is still alive then
he probably has Goering assassinated at some point and replaces him as Fuhrer, simply put he could boss around others and had the muscle to back it up, as well as control of the Gestapo.
The only thing the two of them could probably agree on is that Martin Bohrman's life will be measured in
hours if not minutes before Hitler's corpse is cold they both hated that man so much.
It was actually even worse than that.
At least the Empire had one military force (the EU makes pretty clear that any other paramilitary forces that might have existed were small fry next to the fleet and its stormtroopers). Nazi Germany had at least three distinct militaries: the Heer/Kriegsmarine, the Luftwaffe (worth remembering it had armored and infantry divisions in its ORBAT as well as all the airborne troops and nearly all the heavy artillery), the SS, and an arguable fourth with the Hitler Youth and SA if they became big again once older men demobilized. That one probably would have belonged to Goebbels. All of their leaders hated each others’ guts, had no conscience or checks on them other than Hitler, and were made to compete with each other, sometimes to the point of armed conflict (see Night of the Long Knives), to prevent coups. And there was no real plan for who would succeed Hitler when he died. Yeah...
Add to this the multiple independent fascist or other fellow traveler states that were allied with them like Italy, Romania, Hungary, Bulgaria, and Finland, all of whom would have been scared utterly to death when they got wind of Generalplan Ost and who would have had their own agendas.
Finally, Nazi wunderwaffen were mostly garbage and the headshed’s gutting of their own scientific sector and political hamstringing of their military would have put them far behind the western allies.
Just an utter soup sandwich.
Well that's not completely accurate. The regular military (Heer/Kriegsmarine) was going to most likely be superseded by a more "politically reliable" force if the Nazis won, with soldiers being given the opportunity to continue serving so long as they joined the SS, and the SS expanded from its historic 38 to something like 100 Divisions or so, while the Luftwaffe might keep the (18?) field formations they formed or expand them when Goering becomes Fuhrer. The general staff will be put out to pasture. However, the Kriegsmarine will probably be the only one that survives intact (absent the planned carriers which from what I know of Goering would be a clusterfuck of the highest order).
The SS was the second strongest (and arguably second best, with some elite units) in the whole Nazi military structure. The Luftwaffe ground formations were nowhere near up to scratch, so its not much of a contest as to who was the second most powerful. The Hitler Youth was not a military organ per-se, but would simply end up as a recruiting ground for the SS.
Essentially the SS
was going to become the Nazi military machine since it was loyal to the Party and had no unfortunate memories of the old Germany to fall back on or any moral scruples. The other organs of the party never held a candle to its raw strength, which was why Himmler got the SA gutted, and they would never be coming back on his watch, or Goering's.