Leaving aside the utter impossibility of arriving at this situation in the first place:
Where does the Reich get all this oil from? Because the Caucasus is easily in range of Allied air bases.
How does the Reich get access to all of the UK tech? Are we to assume the British make no effort, at least, to destroy whatever papers/designs they had? They simply hand everything over?
Where does Heisenberg get the necessary stuff needed to make the joke that was the Uranprojekt to churn out A-Bombs?
How does the Reich prevent Allied occupation of North Africa, and a subsequent push from there?
How does the Reich out-produce the US in terms of aircraft, to prevent a 'respectable air campaign'?
With all due respect, I find the notion that the Reich could effectively withstand the US and the remnant British Empire to be implausible, even giving them the British Isles and Russia for free.
Allied naval superiority will allow them to land in whatever peripheral point of their choosing, and starting in October '45, the Americans will be deploying 2-3 bombs per month, and there will be little the Luftwaffe could do to stop even a significant fraction of them.
I agree that the scenario is one of those
justthisclose to ASB ideas, however, if the scenario is accepted as put forward this issue is not quite as you present.
The Caucasus are not within easy strike range of any realistic Allied air base. It is fair to assume that a dismembered USSR is going to be severely restricted by any Treaty and will have a Reich that will be prepared to react with massive force to any attempt to climb off the mat (if there is one country on Earth that understands the danger of allowing a defeated foe to rebuild it is Nazi Germany). If the British are occupied the Italians have either taken Egypt as part of their rebuilt Roman Empire, or the Egyptians have thrown off the British yoke (Even if the British manage to hold on to Egypt somehow the round trip to Baku is 2,600 miles, until the B-29 enters service the Allies can't touch it, and the bombers will be unescorted.). Iraq fought a war with the British IOTL, one that they lost. Not going to lose it this time, not with the Britain occupied.. That effectively eliminates the best jumping off point for any U.S. attempt to move into the Iran. I would say that even continued control of the Raj is more than slightly questionable given the facts on the ground.
The UK might be able to get some of their most valuable people out, but it won't be all of them. Rolls Royce will not have its entire engineering department transferred
en masse, there will be a large number of the designers, draftsmen, etc. who don't get out. Same situation exists for the rest of the British designers. Some of the people who are left behind will have the guts to tell the Nazis to go screw themselves, they will be the distinct minority and one thing that the Gestapo and SS were REALLY good at was making examples of people in motivate others. The British are not going to manage to pull out every physicist in time (If the Heer manages to get ashore it isn't going to take that long to wipe out the Home Guard, the British will be lucky to get the Royal Family and part of the Government out).
A better question than where does Heisenberg get the stuff needed, where does Manhattan? Did the British realize that implosion hydrodynamics expertise would be required once an implosion weapon was determined to be necessary long before the unsuitability of a plutonium based gun-type device was even know? If not, James Tuck, the genius who developed the explosive lensing system that made implosion very possible never left the UK (IOTL he didn't arrive in the U.S. until 1944). Same goes for Sir Geoffrey Taylor, who was also critical to the development of the implosion process. He was an expert in fluid dynamics, hardly the first person that would be evacuated for work on the Bomb. The number of Uranium bombs that can be produced is quite low (it took a full year to enrich enough uranium for Little Boy), so there is not prospect for mass production. At best Manhattan can produce one Little Boy bomb every four months.
North Africa was already touched on above, but with the British out of the war there is little to prospect of a successful invasion of North Africa before 1944, very possibly 1945 or later. Operation Torch was only possible because the Vichy French had minimal resources (and lacked support from a noteworthy segment of the French population). That is unlikely to be the case if the Italians have followed up on their reestablished Roman Empire scheme (which was close to Mussolini's heart). It is also very likely that the Wehrmacht would have a substantial presence on Gibraltar, even if the Rock has been returned to Spain. Such a force would make any effort in North Africa quite difficult.
The Reich now has the enormous resources of the USSR (at a minimum the European portion, although it is likely that "reparations" would be required from the rump USSR as well, to prevent the Soviets from reconstituting if for no other reason). The Soviets have considerable raw materials, factories, and a fully conquered European USSR, coupled with General Government, provided more than enough strategic depth to place factories well beyond the range of any American bomber force (in practical terms the earliest any effort to strike into General Government or further east would not begin unto 1949 at the earliest, when the B-36B (a rather poor platform for the sort of attacks necessary, as mentioned in an earlier post) is in full squadron service. Even then any strikes will be entirely unescorted (even the longest range U.S. fighter, a version of the P-82, only has a combat radius of ~1,100 miles, Berlin is 1,500 miles from the part of Iceland suitable for construction of major air base facilities).
If the UK is gone, the U.S. and Commonwealth have almost no way of getting at the Reich.