Could the Allies have won World War 2 without America?
As usual, It Depends. Assuming that the United States avoided direct declared war with the Axis powers somehow, there are two major scenarios:
(1) The U.S. continues its behavior similar to that in 1940 - i.e. continue Lend-Lease loans and equipment sales to the Allies, and continues to use diplomatic means to hamper the Axis.
Under this scenario, Great Britain likely would have survived further German attacks. However, it is highly unlikely that they would have retained any Asian territories, as Japan would have been able to capture and hold all such territory with little effective opposition (especially not having to contend with the U.S. Navy). The North African campaign would likely ended up as a stalemate, with the U.K. retaining Egypt but little else. Similarly, the U.S.S.R. would almost certainly have continued resistance, and it's successful relocation of industry to the Urals would have provided it with sufficient output to continue resistance.
The likely outcome in this scenario is a negotiated peace between Great Britain and German/Japan, leaving the British Empire smaller but still intact. Such a peace treaty would likely have been signed in 1942 or 1943, at the latest.
A protracted war between the U.S.S.R. and Germany seems most likely, probably ending with a negotiated peace with Germany retaining all of Poland (and, depending on how long the war lasted, some of the Ukraine/Baltic states). Given increasing Soviet industrial capacities, it is highly unlikely that the mis-managed German war industry could have produced enough material to overwhelm the U.S.S.R. Japanese reluctance to engage the U.S.S.R. would have left only a single conflict in Western Russia. Enormous manpower reserves and sufficient (though not by much) equipment would have allowed the Soviets to push the Germans slowly out of their country, though at an enormous cost. The likely result would have been a significant increase in Russian casualties from a protracted war. Japan would have retained all of its conquests, and possibly added India.
In this scenario, the Axis wins on points - retaining most of Europe and Eastern Asia, but the death toll (mostly due to continued Soviet/Russian conflict) would likely have been even greater than actual WW2 figures.
(2) The U.S. reverts to true neutrality, aiding neither side.
In this case, the U.K. almost certainly would sue for peace by 1941, to avoid being starved out by the U-boat attacks. A peace treaty would have been favored by Hitler, stripping the U.K. of most of the British Empire - thus, Germany would have ended up with practically all of British possessions in Africa (most critically, Egypt and the Suez, plus the Arabian oil fields), while Japan takes all of British Asia and likely India (though probably not Australia).
With the added material resources of the former British Empire, and a "winner's" glow, several other (formerly neutral) countries would likely have thrown in on the Axis side, in what would have become labeled a "anti-Bolshevik" crusade. Russia is unlikely to have been able to survive a multi-front war (assuming the Japanese would now enter the war against the U.S.S.R.), and a bolstered Wehrmacht would almost certainly have been able to retain all of Western Russia.
The result here would be a clear Axis victory, with the end of the British Empire, the partition of Russia, and an expansive German and Japanese empires.
The critical contribution of the U.S.A. in 1941-42 is a continued modest stream of equipment to both Russia and Great Britain, plus the morale boost of "help is on the way". This leads to the main contribution of the U.S.A. in both 1943 and 1944 of vast quantities of war material (and significant, though not overwhelming, manpower), and critical technological support. Essentially, the U.S.A. provides a lifeline to allow both Russia and the U.K. to continue to fight in 1941-42, which in turn enables the Allies to stall for time while bringing on-line their vastly superior industrial and technological advantages. WW2 as an "attrition" war heavily favors the Allies, while WW2 as a "fast" war heavily favors the Axis. Keeping the U.S.A. out of WW2 is the critical factor in deciding whether WW2 would be an "attrition" or "blitzkrieg" war.
The British Commonwealth forces in North Africa would have still been victorious without American military aid, they won both battles of El Alamein without the aid of US forces and chased the Afrika Korps back to Tunisia. In Tunisia the combined US and British force tied up part of the newly landed Panzerarmee Afrika whilst the Eight army pursued the remains of the Afrika Korps. This doesn't mean to say that the US help was necessarily required, in fact the British could have just as easily used an entirely British or Commonwealth force for this role and they may have fared just as well or maybe even done more.
This would have been followed by an invasion of Sicily and Italy, as this was the British plan even before US forces were involved. Sicily would have gone very much the same (apart from a famous American General advancing his army through practically empty country, while his allies fought the Germans on a mountain, then claimed to have done all the work).
Italy would have been different though as, with no invasion of Normandy on the cards, it would have made up the chief front for British and commonwealth forces. With all their resources aimed at Italy it has to be said that the allies would have reached Rome a lot sooner.
This no doubt would have been followed by a bloody but no doubt successful invasion of Greece, which would have forced Germany's Balkan allies to divert troops from the eastern front or maybe an invasion of Norway, which could cut off Germany's supply of Iron ore.
All this is dependent on Lend Lease, of course. But let's remember that Lend Lease was a trade agreement that existed before the US entered the war. I mean, the US were selling airplane parts to Germany while they were selling Shermans to the British. [Editor's Note: no, the US was not selling any military parts to Germany post 1939] The lack of lend lease tanks would have prompted British Commonwealth forces to use Churchills and Cromwells more often (and no doubt would have speeded up development of the Comet and Centurian), however the loss of Lend Lease food stuffs would have been a huge factor for the civilian population. But that does not automatically mean British surrender. People prove themselves to be more durable and resourceful than you could imagine, in times of crisis.
All in all, without the US, the British would have been effectively relegated to a support role for the Soviets but between Britain, it's Commonwealth and Russia the allies would have no doubt been successful in Europe. (Asia would have been far more complex and I think I've typed enough)
As the author of the first section here, I would like to dispute the second answer that the British could have survived and even won without more substantial aid than Lend-Lease.
One has to remember several key factors here:
(1) The UK had to import about 25% of its minimum foodstuff requirements - any serious reduction in food imports due to repeated Uboat sinkings of merchant shipping means that the UK starves relatively quickly (weeks, not months), and there's no way to be "resolute" about this; it's starvation. Brtain was also missing significant amounts of raw materials it needed for war material production. Even with the sale of older US destroyers to the UK in 1940, the UK was seriously deficient in convoy protection, and could NOT build destroyers fast enough to counter Germany's Uboat production. In addition, the UK had effectively NO extra shipbuilding capacity to produce merchant shipping (as it was all taken up by naval construction). So, it relied on "neutral" countries to provide ships for carrying goods to it. The likelihood that such neutral countries (including the US) would continue to allow their ships to sail in British convoys while sustaining serious losses is unlikely. Bottom line here: even with Lend-Lease help, Britain is losing the Battle of the Atlantic in 1939-41. Without the massive influx of US shipbuilding capacity in late 1942 (available only after the US enters the war), Britain starves to death by early 1942 at the latest.
(2) Lend-Lease was a political football in the US - it was hardly universally supported. Given that the UK effectively ran out of hard currency reserves by late 1940 and was purchasing solely on credit afterwards, it is highly unlikely that the US would have continued the program indefinitely, since without direct US involvement, perception would be that Germany was winning, and that affected the likelihood that the UK's credit would be repaid. In essence, there was a limit on British credit, and that limit was directly defined by Britain's war outlook (as seen from the US).
(3) Britain's defense of Egypt placed an enormous strain on the British Empire's available military. About half of the deployable (ie not already engaged in active combat elsewhere) British Empire forces were in Egypt for the defense of the Suez, and the logistics for resupply stretched the Royal Navy severely (look at the RN's losses in the Mediterranean during 1940-41). El Alamein still happens (and is won by the British), but there would have been NO counteroffensive, as there's no equipment to use - the equipment that the UK gets from the US is used for home defense, and resupply of the British Egyptian Army to keep it running and able to defend the Suez.
(4) Without the US Navy, Britain's Far East Empire dies. As it stands in 1941, the Japanese conquer practically everything. Britain has nothing to spare for its defense once local forces are destroyed, and the Australians are helpless without US military equipment. India might even have undergone a successful revolution and gained independence - there certainly was a non-trivial effort by Indian nationalists, one which was a serious concern throughout all of WW2.
(5) If the UK is still just trying to keep alive by winning the Battle of the Atlantic all on its own, there's absolutely nothing to support the idea that they would have the excess capacity to build, train, and conduct the North African invasion in 1943, let alone conduct invasions of Italy, the Balkans, or southern France.
Lend-Lease was a lifeline, and nothing more. In essence, it was life-support for the British Empire in 1940-41 - the bare minimum necessary to keep the Empire alive and fighting, but nothing more. It was not enough to provide for any sort of offensive operations. The difference in such an effort (i.e. between manufacturing under Lend-Lease and under US active war participation) is evident when one looks at the difference between 1941 and 1942 US war material production: in virtually all categories (from raw material production, to number of finished items such as ships/tanks/guns, to food and clothes, etc) there is 100 to 1000-fold increase, and the US goes from producing about 5% of the TOTAL world's military equipment in 1939 to almost 60% by 1945.
Today, it's hard to image the difference in production that the US made in 1942 after it became actively involved. To cite just one example: airplane production changed from about 500/month in 1940 to 4,000/month in 1942 to 7,000/month in 1943.