Basically from the beginning. The authorship of the term in its modern sense is attributed to Eric Arthur Blair (better known as George Orwell). In a article 'You and the Atom Bomb' published on 19th October 1945 in 'Tribune', he wrote:

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.

He also used the term next year in the article in 'The Observer' published on 10 March 1946, where he wrote:

[...]after the Moscow conference last December, Russia began to make a 'cold war' on Britain and the British Empire.

As seen above, he uses the term within quoting marks, what strongly suggest that the term was not in popular use back then and the author could have considered it a metaphor of his own invention. The name did not catch on initially, until two years later, when it was used by US presidential advisor Bernard Baruch on 16th April 1947, where is said:

'Let us not be deceived: we are today in the midst of a cold war.'.

This was picked up by Walter Lippman, an American journalist, in a series of essays concerning political relations between US and USSR (later collected in a book 'The Cold War: A Study in US Foreign Policy'), attributing it to the French term 'la guerre froide' referring to the aggressive politics of the Third Reich, including disregard for the peace treaty, Anschluss of Austria and Munich Treaty. It is likely that because of that, even though the term was first used in British newspapers, 'cold war' was considered an American expression, as evidenced by the record of a House of Commons sitting on 22nd January 1948, where Anthony Nutting, MP for Melton said the following:

The time is long overdue for the British Government to abandon this aimless drift; they should recognise that the "cold war," as the Americans call it, is on in earnest that the third world war has, in fact, begun.

It is worthy of note that during the same sitting, the term has been used numerous times, mainly by Konni Zilliacus, MP for Gateshead, who used in social and political context rather than military one, to wit:

I think a somewhat less tendentious description would be to say that it is a policy of containment and cold war against the forces of social change and social revolution in Asia and Europe, which are identified with Communism, which, in its turn, is identified with Soviet aggression and expansion.

and later:

In four months, they have cut it [trade] down from 30 million dollars to 3 million dollars per month, purely as part of their political cold war.

and finally:

The House may think that my judgment of American foreign policy is harsh. If so, I would refer hon. Members to a sympathetic account of it, which was given by one of the most intelligent and best informed of American newspaper publicists, Mr. Walter Lippman, in his book on American foreign policy entitled "The Cold War." He is no starry-eyed idealist, but a hard-boiled realist, who himself accepts the principle of containment and cold war, but objects to the way in which the Administration is trying to carry it out.