I see the Waffen-SS growing and perhaps being somewhat better equipped man for man then the HEER but not replacing it. I can't see it having it's own separate air force or navy, although if the HEER has some aviation units (helicopters etc) the SS will have them too. I can't see a victorious Nazi military with the Luftwaffe having ground formations other than security troops (the equivalent of USAF security forces or the RAF regiment), and control of anti-aircraft forces except those organic to the ground forces. The Kriegsmarine is unlikely to see any significant changes other than those produced by technology. The key there will be what happens when aircraft carriers are in commission, and also when helicopters go to see. Will all aviation be under Goering/Luftwaffe (we know how well that worked for the RN aviation)? Will everything at sea be Kriegsmarine, but land based maritime patrol and attack Luftwaffe (like RAF Coastal Command), or will there be a full spectrum naval aviation like the USN? IMHO the RN/RAF divide is most likely.
Since some people here are suggesting the Waffen-SS would evolve into something similar to OTL Iran's Islamic Revolutionary Guards Corps, I would like to point out that the IRGC has its own air force and navy. Granted, they don't have combat aircraft (just helicopters, jet trainers, transports, and utility aircraft), and their navy mainly consists of fast attack boats, but they have such forces nonetheless. The Waffen-SS would probably be similar. Its air force would be similar in composition to army aviation, while its navy would consist of attack vessels and perhaps patrol vessels for coastal defense.
The Luftwaffe might retain control of the Fallschirmjäger since they're airborne troops and would be lifted into battle on Luftwaffe transports. But I could see the Heer taking control of them after Göring dies.
Regarding naval aviation, I also think a scenario like the OTL Royal Navy and Royal Air Force is most likely: the Kriegsmarine controls sea-based air assets, while the Luftwaffe controls land-based air assets. Göring might try to keep control of naval aviation, but as with the Fallschirmjäger, after he dies then those forces will be transferred to the appropriate branches of the Wehrmacht.
Since the missile program and the nascent nuclear program were more under army control than anything the odds are long range missiles and nuclear weapons would be under army control. I expect the SS will attempt, probably successfully, to be the key link in the nuclear control chain with an SS officer at least involved in nuclear release at all levels, if not physical security alone or with army personnel for nukes.
I think Nazi Germany would deploy a nuclear triad, with each branch of the Wehrmacht assigned a particular leg of the triad. The Heer would control the land-based missiles, everything from ICBMs to tactical missiles. The Kriegsmarine would deploy sea-based ballistic and cruise missiles from submarines and surface ships. The Luftwaffe would have a prized fleet of nuclear-capable bomber aircraft. Each branch would also deploy tactical nukes: nuclear artillery, landmines, and MLRS systems for the Heer; nuclear torpedoes and depth charges for the Kriegsmarine; and nuclear gravity bombs and tactical air-to-ground missiles for the Luftwaffe. The Wehrmacht may even deploy nuclear-armed SAMs and air-to-air missiles. I wouldn't be surprised to see them develop suitcase nukes for use by the Brandenburgers and other special forces units on the battlefield.
The SS would absolutely demand a say in the nuclear authorization process. For example, SS officers would be assigned to the crews of ballistic missile submarines, just like how political officers on OTL Soviet Navy submarines had to give approval for the use of nuclear weapons in addition to the captain.