As of 1940, Germany had a number of severe disadvantages. The first and foremost is that the U.S. had already committed itself to the Allied war effort. Even though the U.S. was technically at peace with the Axis, it had committed itself to sending large numbers of military aircraft to Britain for use against Germany. In 1941, the U.S. produced 19,000 military aircraft, compared to 12,000 for Germany and 5,000 for Japan. The U.S. outproduced both major Axis nations combined even though it was still (technically) at peace until December of that year. Plans were being implemented to expand American military aircraft production to 70,000 per year, with half the produced planes being sent to Britain for use against Germany.
Regardless of whether Hitler did or did not declare war on the U.S., he would still have to face America’s industrial might; and would have to find an answer to the U.S.-built aircraft that would be used to attack Germany’s cities and its people. In addition, the British had imposed a food blockade on Germany, which meant that over the long run there would not be enough food to feed the people within Germany’s borders. Some would have to starve or otherwise be exterminated.
Russia had fought poorly in WWI. It had also fought poorly in the war in 1905 between itself and Japan. Further, the Soviet Union had fought very badly in its invasion of Finland (in 1940). That track record had led German military planners to anticipate it would fight badly if it was invaded. They also believed the Red Army consisted of 200 divisions. (As compared to 150 divisions for the German Army.) This was a severe underestimate: by the fall of '41, the Red Army consisted of 600 divisions. Likewise, German military planners had grossly underestimated the industrial capacity of the Soviet Union.
Germany’s supply lines were a problem due to its lack of oil. In the long term that problem could be solved by upgrading its rail network and using coal-powered trains to move supplies. The plan was to surround and capture the bulk of the Red Army in the western regions most accessible to the German Army. With the core of the Soviet military strength broken, the German Army could then push eastwards to take the grain, oil, industrial capacity, and manpower it so desperately needed. These resources would allow it to counter the large numbers of British- and American-made aircraft being used against Germany.
Over the short-term, however, the invasion would make Germany’s food situation worse. Most western Soviet territories ran at a food deficit. Even the Ukraine’s food surplus was not as large as it once had been due to Stalin’s industrialization and collectivization efforts. Because Germany didn’t have the food it required to feed everyone, the solution was to starve or exterminate those it liked the least. Jews received the lowest priority for food allocations, and people in the occupied Soviet territories the second-lowest. Because the British (and later American) food blockade had forced Germany into a situation where it had to starve people in its occupied territories, Hitler felt that getting local populations to cooperate through fear might be more effective than would a more positive approach. It’s difficult to convince a population that you are their friend when you are starving them to death. Germany had no food with which to avert widespread starvation.
Clearly, the invasion of the Soviet Union failed to solve Germany’s problems. In fact it greatly added to them. Over the short-term at least, Stalin would likely have been content to sit and watch Germany and the Western democracies fight each other. He regarded both sides as equally enemies, and wanted nothing more than a long, bloody war which would bleed both sides white. Once Europe was sufficiently weakened and war-weary, the Red Army would of course move west to pick up the pieces. Stalin’s approach meant that over the short-term at least, Germany could avoid facing the Red Army if it wished to do so. (Though it would have to face that army eventually.) However, the Red Army had experienced a purge several years ago, in order to eliminate the old, gentlemanly officer corps, and to replace it with one more loyal to communism. (I’ve seen it alleged that the new officer corps was intended to be more thuggish–an accusation which seems borne out by the atrocities committed by the Red Army.) Also, in the spring of 1941 the Red Army was in the midst of a doctrine change. With the Red Army still recovering from the purge, and in the midst of a doctrine change, there was an opportunity to attack it when it was unready–an opportunity which would not have existed in '43 or '44.
In the U.S., some media outlets were isolationist, while others were interventionist. However, the isolationist media outlets were being bought up by those who favored interventionism. The American public had recognized the U.S. had been duped into entering into WWI, and that sentiment made it reluctant to enter another European war. But with American media outlets and the president being strongly pro-war (and generally pro-communist), Hitler felt it was only a matter of time before the U.S. declared war on Germany. (Much like it had during WWI.) He also mistakenly believed that the U.S. would initially be too occupied with its war against Japan to do much against Germany. Declaring war meant that Germany would have to deal with the American Army + its military production, instead of just its military production. Whether that would or wouldn’t have mattered over the long-term depends on how successful FDR and his allies in the media would have been in getting the U.S. into the war had Germany not declared war.
It is also worth noting that by 1941 the U.S. had broken Japan’s diplomatic code, which meant that the U.S. government knew more about the goings-on in Tokyo than did the Japanese ambassador. Specifically, they knew that if they presented a reasonable offer for lifting the oil embargo Japan would accept. But if the U.S. asked for dramatic concessions on Japan’s part, Japan would go to war within a matter of weeks. The U.S. asked for very dramatic concessions indeed. The war between the U.S. and Japan helped turn the latter nation’s attention away from the Soviet Union; while also giving FDR the chance he needed to fully mobilize America’s resources for war. Also, a leaked U.S. government document persuaded Hitler that the U.S. was temporarily too weak for a two ocean war; and that he could therefore get away with sinking the massive quantities of Lend-Lease Aid flooding into the Soviet Union and Britain. Shortly after the Japanese attack on Pearl Harbor, Stalin shipped 100 divisions west across the Trans-Siberian railway because he knew Japan could no longer seriously attack him. Those 100 divisions arrived in the dead of winter, and proved critical to the Soviets’ success.
To make a long story short, Germany in 1940 had very few good options. Germany was weaker than either the U.S. or the U.S.S.R., and the leaders of both nations were committed to the destruction of the German military. The case could be made that Germany had significantly less long-term strength than did Britain–a nation which was already at war against Germany in 1940. The conquest of North Africa and the Middle East would have helped Germany’s situation somewhat, especially in terms of oil. Had Hitler chosen that path, Germany’s situation would have improved with respect to Britain. Also, the delay would have allowed Germany to build up its own industry. However, the U.S. and the Soviet Union would also have become significantly stronger during the delay.
The only chance for an Axis victory I can see would have been under the following scenario. In 1941 Germany invades Turkey and North Africa; and secures control over the Middle East. It begins recruiting large numbers of Muslim men for its army, which it sends against the British force in India. Then in the winter of '41, Japan declares war against the U.K., but not the U.S. Germany and Japan would combine to take India, and would divide it between the two of them. Germany would then dedicate 1942 to improving its industrial capacity, developing new, more modern weapons, fending off Allied bombing raids, submarine warfare, and southward conquest along the Nile. It would also attempt to recruit soldiers from India’s Muslim population.
1943 would likely be similar to '42. The main difference would be that there would be significantly more American-made planes being shipped to Britain every year. Germany would continue to push south in Africa, with the intended goal of taking South Africa. If Africa fell to the Axis, and if Australia and New Zealand negotiated a separate peace treaty with Japan, the British war effort would then be limited to the British Isles and Canada.
In 1944 Hitler would invade the Soviet Union. The Red Army would have been significantly stronger in '44 than it had been in '41. But several factors could offset that. 1) Germany could invade from the south (Persia), in addition to from the west. 2) Germany would have a much larger initial invasion force because of help from the Muslim men it had recruited. 3) During WWII, Germany massively expanded its military production between '42 and '44. It produced nearly three times as many military aircraft in '44 as in '42; and four times as many tanks. This scenario assumes that a somewhat similar production increase would have taken place. 4) Later in the war, Germany was significantly ahead of its enemies in most major technological areas. Its Me 262 jets achieved a 5:1 kill ratio; and that ratio increased to 10:1 when they were equipped with its best air-to-air missiles. Later in the war it created the assault rifle, it had the best tank designs, it had the best handheld anti-tank weapons, its Type XXI U-boats were the best submarines of the war, etc. Assuming the critical years of the war between Germany and the Soviet Union would have been '44 and '45, many of these technological advances could have appeared in sufficient numbers to make a difference. (Especially if that had been the plan from '40 onward.)
Under this scenario, the plan would not necessarily have been to conquer the Soviet Union in its entirety. Instead, Germany would conquer whatever it could gain during two years of conquest. After that it would negotiate a peace treaty with Stalin. Conquest of a significant portion of Soviet lands–ideally including Moscow, the Caucasus oil fields, and other strategically important areas–would provide Hitler with the resources he needed to hold his own in the long, grinding air and sea war against Britain. FDR would die in '45. It’s likely that his successor would either have been less pro-communist/committed to Germany’s destruction than FDR had been, or else would lack FDR’s ability to get Congress to do his bidding. Further, Britain would be in dire financial straits by this point, making it difficult to justify further increases in Lend-Lease Aid. (The U.K. was close to bankruptcy even as early as '40.) With its colonies conquered and with Germany victorious over the Soviet Union, and (possibly) with the American commitment to Lend-Lease Aid lessening, there is at least the chance that Churchill might have been replaced with some other leader more willing to negotiate a peace treaty. That peace treaty would have ended the war at last, and would have allowed Germany to escape the horror of postwar Soviet occupation.
On the other hand, it’s quite possible the British wouldn’t have agreed to peace; and that the Americans would have become increasingly pro-war due to the institutional influences described earlier. If those things were to occur, then in 1946 Germany’s plan would have been to focus on building Type XXI U-boats to sink much of the British Navy, while using its jets to gain control over the skies above Britain and Germany. Naval and air superiority would pave the way for an invasion of the U.K. in late '46 or early '47. At that point, Germany could at last achieve peace, unless of course the U.S. had declared war on it. But by this point, Germany would have the industrial capacity, food supplies, oil, and access to raw materials necessary to hold its own even against the U.S.
I don’t know whether the above plan would have worked. But, at least with 20/20 hindsight, this seems to have been among the best available plans to ensure an Axis victory.