I don’t believe that Churchill, or FDR/Truman would have ever seriously considered deliberately starting a war with the Soviet Union. It is true that the USSR was an evil state, and was rapidly replacing Nazi Germany as the threat, but – IIRC – a lot of the planning concluded with ‘do not do this cool thing:’ the war would be immensely costly even if the allies won, and Britain was in no state to fight and win a war with the Russians. The state was already bankrupt. Even if Churchill thought it was better get the war over with, he isn’t going to be in power much longer.
A more reasonable prospect is that Stalin hears from his spies about Operation Unthinkable and assumes that Churchill and Truman actually intend to go through with it. He recognised that Churchill had a spine, and that he was deeply concerned about the communists, and feared that Truman might be cut from the same cloth. In this state, he might think the plans are genuine, in the sense the politicians think they can be safely implemented and lead to victory. He would also be very aware of the atomic bomb, and that while production was currently very slow it would not be long before America managed to start churning out 50 or 60 bombs a year. The Russians had an atomic programme of their own, and it might make progress, but there was no guarantee that the Russia would get the bomb in time to deter an American/British attack.
If we run with this, Stalin might decide that his best chance to fight and win a war is to strike, as hard as possible. This would represent the only realistic chance for Russia to win the war. If they managed to drive the British and Americans back to Britain, and suppress the French with the aid of the French Communists, there would be enough space between American airbases and Russia for the atomic bomb to become much less of an issue. Stalin could also invade out of Russian controlled-Iran, threatening the remnants of the British Empire in India and intimidating the Turks into submission. It would look good, at least on paper.
The real world might not be so obliging. At first, the Red Army would have a major advantage. Their tanks are better, and they have mastered the art of mass movement. The allies have also started moving troops to the Far East to prepare for a invasion of Japan. On the other hand, Allied air power would be vastly superior to its Russian counterpart and it would not take long before Russian supply lines were being bombed heavily and the spearheads started to run out of fuel and ammunition. Various communist factions would cause havoc in the rear, but they would probably lose popular support very quickly as word of Russian atrocities spread west. The Russians themselves would not be particular enthusiastic about the war. They hated and feared Hitler, with good reason, but they have nothing in particular against the British and Americans. Pro-communist propaganda was nowhere near as successful as the atrocities committed by the Germans, when it came to winding up the enthusiasm for war. There’s also the simple logistics issue of getting from Berlin to Normandy, a distance big enough to make it quite possible for the allies to put together defensive lines before the Russian offenses ran out of steam.
If the Russians do crack the defences and conquer most of Western Europe, they might have a chance to win. If nothing else, the allies have to launch a rerun of D-day as well as dropping atomic bombs on European targets to clear the way. I don’t believe they could succeed before their economy starts suffering from the sudden loss of lend lease. They would have major shortages very quickly, particularly when – if - American bombers start striking the oil wells at Baku. Their army would slow down rapidly, perhaps even grind to a halt. The allies could then start mounting a counteroffensive, which would push further and further into the USSR. Unlike Hitler, the British and Americans and French would treat the locals decently and work hard to support anti-Communist factions within the USSR. At some point, the Americans would be able to fly bombers over Moscow and start nuking cities within the USSR (although targeting industrial plants might be more useful, in the short-term.) Taking out Moscow would do immense damage to the soviet union, if only because it would kill a vast number of the bureaucrats who keep the system running.
I can turn this into a full scenario, if anyone’s interested.