I don't think the Red Army would be able to invade Iran. The Zagros mountains would provide a very defensive posture for British and Commonwealth forces stationed there.
They did historically, and it was during the first months of Operation Barbarossa too. They pretty much crushed the Iranians and secured a lend-lease line quickly. Of course it would be the opposite objective but I can imagine that it would be roughly the same route and timeframe, but with far less pressure and probably better equipped units (because all priority went to the Soviet front).