Well the implication of a Cold War is that it is well... cold, a hot war would be the West and the Soviet Union slugging it out for Europe, like imp said nobody was about ready to do that because everyone was afraid of the big nuclear trump card that both sides possessed.
So we had the Cold War, where two sides that obviously were in conflict found different ways to fight each other. It was less the time of the soldier and the pilot and more the time of the spy and the diplomat. Instead of fighting the US in the Middle East the Soviet Union backed the enemies of US allies, namely the Arab states, against Israel. The US backed Israel and intervened in several Middle Eastern nations to ensure cooperative pro-US governments, which met with limited and mostly temporary success followed by some catastrophically bad failures.
Your proposal's kind of an early version of Operation Unthinkable, a plan Churchill developed to invade the Soviet Union. However an AH Cold War would perhaps involve a three-way scramble between Nazi Germany, the West, and the Soviet Union for hegemony in Europe. Again, nothing like a nuclear trump card to keep all three groups afraid.
Edit: Also, the historical Cold War was because of a falling-out between the USA and the USSR after the alliance of convenience that they made in World War Two. An earlier split would have to be so early as to very possibly butterfly the Soviet Union's existence, given that...
a. WWI and Czar Nicholas' incompetent leadership during said war lead to the Russian Revolution, none of the issues with the Russian monarchy were new, it's just that Nick provided the catalyst by dragging Russia into a massive war and promptly got his ass handed to him trying to use the old Russian steamroller. An earlier falling-out between the Western powers and Russia may have involved a profoundly different WWI (i.e. possibility of Russia reluctantly joining the Central Powers out of necessity despite Austria-Hungary's involvement).
b. Britain and France would've probably dealt with Hitler first anyway. For Britain the Soviet Union was a threat, but a relatively distant one, it didn't have the naval power as of yet to challenge Britain. Germany was a threat looming right on their doorstep (it could take Belgium and springboard into Britain, which is historically a reason for the British to fight on behalf of Belgian neutrality as was policy of keeping any one power from majorly consolidating in the continent), for France it was even more so. Attacking the USSR while leaving Germany alone would be ignoring a huge elephant in the room and opening them up for the possibility of the Germans forming an opportunistic alliance with the Soviet Union to gain support against Britain and France.
c. Nazi Germany never had the capacity to fully conquer Russia. Steamroll their armies and extract a humiliating peace from the defeated Soviets but never occupy them, even with British help any long-term holding of significant portions of Soviet territory is difficult to conceive.