That would only happen if Stalin was incredibly desperate. I believe plans were made for British forces in the Middle East to move into the Caucasus during the worst days of Case Blue, but it was never necessary. Stalin did not want foreigners to see what life was like in the Soviet Union on their own. Military units couldn't be controlled by handlers like diplomats and reporters were.
There'd need to be a complete disaster on the Eastern Front where Western forces had to plug in gaps while the Soviets pulled up and trained new reserves. For Western forces to be permanently deployed in the Eastern Front (not just for a year long period until Soviets could bring up new troops), we are looking at a huge disaster that has to prevent the large Soviet offensives that marked the 1943-1945 period. If they can't man their entire front, they really can't launch deep operations.
So any scenario where Western forces are allowed by Stalin on the Eastern Front is a scenario where the Soviets have their advances greatly limited. And Allied forces would probably have their logistical trail in the south through the Middle East so they'd be heading along the Black Sea into Ukraine and finally the Balkans. Regardless if the Western Allies concentrate on a Western Front, or supporting the Eastern front, the postwar period will be very different. Not only would Romania and Hungary be much more willing to surrender to American and British forces, their occupation would be very different.
So what if we have a scenario where Allied forces only temporarily enter the Soviet Union to protect the Caucasus after some 1941-1942 disaster where the Soviets can recover from 1943? Afterwards, the Allies leave. Well, it means the 1943 frontlines have to be far worse so the Red Army will be delayed entering Eastern Europe. It also means the US and Britain learn a LOT about life in the Soviet Union from their brief occupation of Georgia, Armenia, Azerbaijan, and perhaps the northern Caucasus/Kuban. That will greatly change how the governments of US and Britain will react when it comes to postwar planning. As news gets out as to what people told the Western troops while they were there, it may even cause a huge domestic and international incident as the American and British people react to it.