This thread has become quite vague in its discussion because we do not have a clear POD.
When exactly does Britain stay out of the war? Is it 1940 (after the Battle of France) or is it 1941 (immediately prior to Barbarossa OTLs date) or somewhere in between? Depending on the timetable a lot changes.
Effects with a POD after the British defeat in France:
Italy is at peace with Great Britain, no need for North African Campaign.
Italy still invades Greece and has a) more resources for that theater of operations b) freedom in the Mediterranean. No British aid to Greece. We assume Greece still does not allow the Italians in (questionable, since they would be the only nation facing the Axis, if they decide to fight and wont get British aid).
Greece can still hold off the Italians at the Albanian front for some time and push back, but Italy has the capacity to invade and hold several islands and potentially even Crete, after neutralizing the small Greek fleet. No need for German paratroopers in Crete. The invasion through Albania was hampered by logistics in OTL due to Albanian port capacity, the situation is not better in this ATL. It's possible that Greece will sue for peace at some point however if lots of islands are lost and more pressure is put on by the Italians. The RM can roam freely in the Aegean / Eastern Med without the RN opposing it. Even if the Greeks dont seek truce and the Germans do not intervene, the Italians will prevail at some point. The Germans intervened in OTL not to "rescue the Italians", but to make certain the British did not gain a foothold in the Balkans. With Britain out of the war, they no longer need to do that. The Italians can continue their offensive from Albania, starve off Greece and grab islands, up until the Greeks decide to surrender or crumble. It will be inevitable without British support.
It is questionable whether or not a Yugoslav campaign would be necessary. At this point Germany is the sole ruling power and the Yugoslav officers may decide to call off the coup in fact.
Avoidance of the North African campaign & the Balkan campaign leads to Barbarossa possibly able to start 1-2 weeks earlier (I know airports were not ready and muddy weather still remains, but 1-2 weeks earlier is potentially possible). This means that the Germans may reach Moscow before winter kicks in and Soviet reinforcements arrive. If Stalin is not killed or couped the war will still drag on, since the Germans will still be pushed back, albeit with potentially lower losses. If Stalin is killed or couped, there may be a truce.
But let's assume that either Barbarossa is not launced earlier or that Stalin remains in power and the war continues.
Come 1942, the Germans will be more powerful in the Eastern Front. With Britain out of the war they do not need that many resources allocated to the West, the Med and N. Africa.
In OTL more than 40% of the fighter planes of the Luftwaffe were in the Western Front & the Med in 1942. No need for that any more. The extra couple divisions not in Africa may become a handy reserve force in the Eastern force.
One can also expect the Italians to contribute more forces in mainland Europe, now that they won't suffer all the losses they had in Northern & Eastern Africa, freeing up German troops from anti-partisan and occupation duties.
Turkey will be pushed to enter the war on the Axis side and it may indeed happen. Even if Turkey does not enter the war, the Germans will be faster and more successful in their summer offensive of 1942 in the Soviet Union. They have more trucks (lot's of those went to the Afrika Korps in OTL), they have a lot more planes and they have more troops. If the Turks join the war or open the Straights for the RM to enter the Black Sea, the Russians are in even deeper trouble.
Stalingrad will be captured early and not ignored since the German advance will be swifter in this OTL and still even if the Russians manage to pull off a counteroffensive they won't be as mobile as in OTL due to the lack of trucks and tanks without Lend Lease. They will also need to be more cautious when it comes to munitions expenditures, so Soviet artillery barrages will be lighter.
We are also bound to see a shift in German war production in 1941. With Britain out of the war, priorities will be different. Less steel and industrial capacity will go to flak batteries, fortifications in the West, u-boats and more will go into tanks, artillery and logistics (trains, trucks). I expect the Soviets to lose the Caucasus in late 1942 in this ATL.
Either Stalin throws in the towel after that and seeks truce or the Germans launch a second attempt at Moscow then.
Another factor is what happens with the US in this ATL. I am not sure, if Hitler would declare war as in OTL in 1941 after the Japanese attack. Even if he does declare war, the US are in a tough position on how to harm Germany.
An outright transatlantic invasion in Aftica (transatlantic Torch) is out of the question. With Britain out of the war, it will be very difficult for the US to help out the Soviets, trying to send in convoys to Murmansk would be suicidal.
P.S. One interesting butterfly is that if truce between Germany and Britain is signed immediately after the evacuation of BEF from France, the French fleet may not be attacked and largely neutralized by the RN. This does have potential butterflies, if the Germans manage to grab it at some later timepoint...