They absolutely collaborated during the war, and while it's hard to say that they would have lost or exactly how the war would have been different, it would nevertheless have made the fighting substantially more difficult. I would say it's a misnomer to think of the USSR as separate from the Allies, because Soviet troops are such a critical aspect of the manpower needed to win the war. As early as 1941, even before the United States formally enters the fighting, Roosevelt insists that Lend-Lease supplies be sent to the Soviet Union as well, despite the fears of his advisors that the country will soon collapse.

In terms of war goods and materiel, the Soviet Union is the second-largest recipient of aid from the United States under Lend-Lease (behind Great Britain). In his memoirs, Nikita Khrushchev claimed that Stalin said the war could not have been won without aid from the United States, and Khrushchev endorsed this position as well (Memoirs of Nikita Khrushchev, 1918-1945, Volume 1). Even with the truly heroic efforts in Soviet industry to set up factories in the safe hinterland of the country, it still has tremendous need for just about every war good, on top of more ordinary but still vital supplies: trucks, jeeps, planes, and even food were all critical to keeping the war effort. Remember that with the loss of the Ukraine, the USSR had lost some of its most agriculturally productive lands, and American food kept the Soviet army fed at a time when every able-bodied man who could be diverted to the fighting was being sent. Richard Overy's Why the Allies Won gives a good sense of the sheer industrial scale that gave the Allies the edge and why their victory was inevitable, but that edge depended on very close collaboration between them.

In turn, the Soviets shoulder the greatest burden of the fighting against Nazi Germany for the majority of the war. Over a million Germans die fighting on the Eastern Front, and another million plus are captured, whereas just over 100,000 are killed fighting in France and Belgium throughout the war. Soviet troops make the victory possible, in part because Britain and the U.S. also are divided in fighting Japan, but mostly because the Soviets are stuck dealing with the German war machine. This produces tensions between the Allies, to be sure, because at Tehran Stalin demands and insists that a second front be opened up as soon as possible to take pressure off of his armies, which is in part why Churchill and Roosevelt are so insistent that maximum logistical support is diverted to the Soviets; they need to keep them in the war, and they worry about the possibility of Stalin concluding a separate peace with Hitler if he felt he didn't have a better option. But it also leads to American and British commitments to open a second front. After the Tehran Conference in 1943, they agree to a May 1944 (postponed to June) invasion of France, and that Stalin would time similar assaults in the East to tie down German forces and stretch them thin.

They also collaborate closely in the planning for a postwar world. Once the U.S. joins the war on top of supplying Britain and Russia, most rational observers understood that Germany had lost; in his memoirs, Churchill made it clear that once he heard the news of Germany's declaration of war, he felt that they had lost and they simply needed to be smashed into submission. As a consequence, they immediately begin planning for what a postwar world is going to look like. Some of that simply relates to occupation of postwar territories, like the division of Germany that's discussed at Yalta, but it also goes beyond. The Soviets participate in the Bretton Woods Conference in 1944 with America and Britain, which was aimed at stabilizing currencies to guarantee trade in the postwar period, which would in turn guarantee prosperity and prevent a recurrence of the war (Roosevelt in particular believed that the collapse of global trade had gone a long way in fueling the rise of dictatorships and the eventual war).

In short, not only did cooperation save lives and shorten the war, but actors on both sides understood they needed the other for the victory they wanted.

Some useful reading... Robert Dallek, Franklin D. Roosevelt and U.S. Foreign Policy Paul Kennedy, The Rise and Fall of the Great Powers Richard Overy, Why the Allies Won