A bit provocative but I think Markin actually makes some good points. I do not think the British could have taken over control of North Africa without American help - especially in three key areas - tanks, aircraft and general logistics.
Tanks in North Africa
Parsifal already outlined the basic situation with the tanks, but let me spell that out a little bit further. British tanks like the Cruiser and Crusader and even the Matilda infantry tank were carrying
either anti-tank or HE weapons. You had a lot with
2 pounder QF (40mm) medium velocity AP guns, and some designated as 'CS' (close support) tanks with
3 inch (75mm) low velocity HE guns. But it was an 'either / or' thing. Later some had the 6 pounder (57mm high velocity) which was a much better AP weapon, but still not very good against non vehicle targets. Most of the British tanks were fast, but also didn't carry a lot of machine guns (usually just one .30 cal weapon) and with the exception of the Matilda had poor armor protection by the standards of the day and a tendency to catch fire when hit.
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Here is the big problem with the British tanks - the 2 pounder and the 6 pounder were both pretty good AP guns for use against tanks - the 2 pounder was roughly equivalent to the American and German 37mm guns, while the 6 pounder was very good at relatively short range. But the 2 pounder had
no HE shell, neither did the 6 pounder - at least not then during the Desert campaign. Conversely, the 3 inch HE gun on the 'CS' tanks had no AP capacity. German defenses in the North African tank battles tended to be a mix of tanks and AT guns, (including the famous 88 but much more significant were smaller 37mm, 50mm and 75mm guns).
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So if a column of Cruiser Mk IV and V or Crusader tanks runs into Panzer Mk III's, they have a reasonable chance of success, but if they are faced at the same time with some well placed German 50mm AT guns, they have really no way to answer back. They can shoot with their 40mm AP gun (2 lber) but AP ammo isn't very effective against enemy gun positions. By the time a CS tank rolls up they are likely to be wiped out. They also don't have any way to silence machine gun positions, mortars, or artillery they encounter.
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By contrast the German tanks by the time of Tobruk are made up mainly of Pz III and IV, StuGG III plus some Italian and captured British tanks, but most of the German tanks are armed with 50mm medium velocity, with some 75mm low velocity and the F2 Special armed with 75mm high velocity guns. All dual purpose and effective against both soft targets (infantry, spotters, artillery or AT guns) and hard targets (tanks and other armored vehicles). Plus usually two fast firing machine guns per tank. The German 50mm gun outranged the 2 pounder and the 75mm outranged the 6 pounder. This was one of the reasons why they usually won out in more or less numerically and situationally equal tank battles against the British. The British are highly vulnerable to German AT guns, don't have enough effective radios, and also lose out in tank vs. tank encounters.
You can really see how this plays out in more realistic tabletop or even computer war games simulating tank battles in this era.
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Enter the US kit. The M3 / Stuart was roughly equivalent in most significant respects to the older British Cruiser tanks, fast and lightly armored with a small high velocity (37mm gun), but it
did have AP ammo for it's small gun (and also very effective cannister shells for short range) and more machine guns and radios. It was never designed to be a medium tank but it was useful for recon and screening etc. But overall no major improvement, incremental at best, over a Cruiser Mk IV and a little less useful than a Crusader.
However the M3 Medium (with both the same 37mm high velocity gun of the Stuart and a 75mm bow gun) was
very useful because of it's medium velocity 75mm. In spite of limited traverse etc., it's gun had good AP and HE rounds and could knock out both German tanks and gun positions fairly efficiently.
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When the M4 (Sherman) showed up, contrary to the tropes it was probably the best tank on the battlefield at the time. It had good and reliable radios. Heavy armor by the standards of the day - more than the German Pz IV. It's medium velocity 75mm gun (a little more powerful than the gun on the M3) had a long range, could kill any German tank of the time except the very rare Tiger, could outrange all the German guns except the 88 and the high velocity 75mm on the (also pretty rare) Pz IV F2 Special and some precious Pak 40 AT guns. could also flatten anti-tank gun positions, spotters and artillery as quickly as they were detected and even had a gyrostabilized gun which could shoot on the move. They also carried multiple machine guns including .50 cal heavy machine guns which were much more effective against light vehicles (including German light tanks and armored cars) and soft targets out to a far greater range than the LMG's.
That also goes for American made self propelled artillery, half-tracks, and even trucks - everything had a .50 caliber machine gun on it. The Germans who fought in the Afrika Korps commented on the .50 cal quite a bit and clearly feared it.
The planes
In the early days in North Africa the British were making do with Lysanders and Gladiators, facing mostly Italian Cr 32 and Cr 42 fighters with a mix of obsolete bombers on both sides. The Germans upped the ante by sending some Bf 110s and the Italians sent more capable MC 200 and Fiat G.50 fighters. The British answered with the Hurricane which had it's day and could handle the opposition at first, but when the Germans sent Bf 109s to the Theater, the Hurricane quickly showed it's limits. The main bombers at the time were Blenheims on the British side and a mix of Ju 87, SM. 79, CANT 1007 and Ju 88s on the German side.
Very quickly from Spring of 1941 it became clear that the Hurricane was no longer in the game. The Blenheim was notoriously ineffective as a bomber in this Theater and crews took even worse casualties than the Hurricane squadrons. US made P-40 Tomahawks and later Kittyhawks may not have turned the tide, but they quickly became the main air superiority type and stabilized the disaster. American made bombers, some coming via the French some directly from Lend Lease, quickly formed the backbone of the British bomber force - Martin Marylands and Baltimores were first, then A-20s, and eventually B-25s and B-26's in American hands.
By the time of El Alamein US made P-40s were definitely taking the brunt of the fighting and shooting down the majority of enemy aircraft, as well as dropping the bombs which directly led to the key breakouts, notably by 112, 250 and 260 Squadrons RAF and the US 57th Fighter Group, while US made medium and heavy bombers had the most impact on enemy airfields, supply and communication assets. Not long after El Alamein by the way the Hurricane was basically retired from the front line even for fighter-bomber missions, they taper off you don't see them in action almost at all after the 1st quarter of 1943.
TL : DR I don't think the British / Commonwealth forces could have won El Alamein without Sherman tanks, US artillery, ammunition, trucks and food and fuel, and the P-40 fighter and a variety of US made bombers. Furthermore I'm not sure the British could have turned the tide in North Africa alone. I definitely don't think they could have captured Sicily and thus secured their Malta supply line alone.
For Barbarossa, it's a bit trickier to say for sure but I know American and British tanks were important in the Soviet Army by 1942 and I know for a fact that US aircraft were important in the defense of Moscow and Leningrad, and to a lesser extent in Stalingrad as well. All of these were basically 'tipping points' in the War.