If the Germans capture Stalingrad and the Caucasian oil fields,than I could see a British bombing campaign in the Caucasus,but sending troops on the Eastern Front would be hard,tough possible,if wanted,also from the North,from convoys landed at Murmansk and Archangelsk,but,leaving aside the,,This man(Russian) is your friend;He fights for freedom" thing,are the British soldiers really going to die for the Soviet Union?I mean,would Churchill send troops to fight for the reestablishment of the Soviet regimes of the Nazi-occupied USSR?And after that,would the British fight for Stalin to regain half of Poland and Bessarabia and then subdue Eastern Europe?And would the British soldiers have nothing to say when they would witness they're Soviet comrades raping German women?Not to take into account the huge casualtie rate of the Eastern Front,which,if the rabidly anti-communist Churchill,won't find useless for British interests,the British people would.Though it would be cool to see a joint Sovieto-British assault on Berlin,the British simply had no reason to fight the Germans through Russia and gain nothing in exchange,when they could fight Germany through North Africa,Italy and France,where they really had interest.It's not like they expected to fight WWIII with the Soviets when they meeted head-on,it's just that they saw the Soviets as the lesser-of-two-evils and they expected to see communism "somehow" dissapearing after the end of the war.Even if the Soviet Union was to be seen as losing the war,none of the main Allied leaders proposed sending an Allied Expeditionary Force.
P.S.:Even in the ASB scenario of an Eastern AEF,Monty would not left the main Western Allied theatre,only to be bossed by Zhukov,as the picture seems to show.