George Orwell first used the term 'cold war' is his essay titled "You and the Atomic Bomb". This essay was written 2 months after the atomic bombs were dropped on Japan and it was published in the British newspaper Tribune.

We may be heading not for general breakdown but for an epoch as horribly stable as the slave empires of antiquity. James Burnham's theory has been much discussed, but few people have yet considered its ideological implications — that is, the kind of world-view, the kind of beliefs, and the social structure that would probably prevail in a state which was at once unconquerable and in a permanent state of ‘cold war’ with its neighbours.

Full essay can be read here - http://orwell.ru/library/articles/ABomb/english/e_abomb

There are two ways this kind of question is usually answered.

The first is the OED-style, "who said it first" approach. No doubt this is of some interest but personally I find it not very useful. I mean, just because the OED found some early reference to a term doesn't mean it was common, and doesn't mean that the reference was influential. The OED, incidentally, lists Orwell as the first usage. This tells you nothing, however, about when it "came into vogue."

The other way is to look at relative usage over time. This has been enhanced greatly with tools like Google's Ngram Viewer, which lets you chart the relative frequency of words across the Google Books corpus. It isn't perfect — they often have their dating wrong and there are plenty of bad OCR transcriptions — but the ease in creating such graphs and their general alignment with our broader, cultural-historical understanding of the period makes them a potent data point. Here is the Google Ngrams result for 'cold war' and 'Cold War'. As you can see, we see a steep rise in the lower-case version of the term from 1945 through 1964, then a supplanting of it by a capitalized version therein.

Unsurprisingly (if you are an historian) people started really talking about the Cold War after it ended — terms for eras are often more useful after the era is over.

The transition from lower-case to capitalized is often important as it can linguistically signal a shift from a term people are using in a merely descriptive or an uncertain way (it is of note that in the 1940s, as well, "cold war" was often in quotation marks), to a term that has been accepted as a legitimate, standard, common one for a given period of time (consider the difference between "middle ages" and "Middle Ages"). By the late 1960s, the "cold war" had ceased to be something that the USA and USSR were engaged in, and instead had become a period of time itself.