A quick answer is that it is a myth, or maybe more of a joke.

The most widely quoted origins is the rather vague “sometime in the 1930’s or 1940’s”, or the much more precise “22nd of October, 1957”. It is more often than not attributed to the Times and occasionally the Mirror.

I found a rather dubious website that has a list of occurrences of this phrase from the early 1930's.

From Kingsley Martin's "The British Press and Foreign Affairs," The Political Quarterly [London], Vol. 2, No. 1, January-March 1931, pp. 115-120. A comparison with the Press of most continental countries would show that we remain a singularly insular people. When, not long ago, a great storm at sea damaged the cables and communication with Europe became for the moment difficult, The Times headline ran "Continent Isolated," and the significant fact about that superb example of sub-editorial genius is that it passed almost without comment from English readers. [p. 115]

From "In the Driftway," The Nation [New York], 22 April 1931, pp. 450-451. ["The Drifter," the author of this piece, refers to himself in the third person.] For the present, however, he will cut himself short with the story of a headline from the London Times. It was told to him only recently by a Scotchman -- if, by the way, you are in search of entertainment, persuade a Scotchman to speak his mind and heart about Englishmen. The headline in question appeared after a severe storm in the Channel which had paralyzed shipping. "Heavy Channel Storms," stated the Times with dignity and without disguise, "Continent Isolated." [p. 451]

From Harold E. Scarborough's _England Muddles Through_ [New York: The Macmillan Company, 1932]. Presumably the London Times saw nothing amusing in its headline, "Continent Isolated," when for three days a storm held up shipping in the Channel. It was an American woman resident in England who called my attention to this headline, and when I showed it to an English journalist he inquired, in honest bewilderment: "Well, what's wrong with it?" [p. 92]

From "London Calm over Decline," Wireless to The New York Times, The New York Times, 25 October 1932, p. 31; ProQuest Historical. The violence of the decline which took the pound to the lowest level since Dec. 2, shook almost every department of the Stock Exchange. Financial writers professed to be undisturbed, however, and the outside public was kept calm by newspaper tactics, recalling the classic headline a London paper once used, "Great Channel Storm -- Continent Isolated.

All of which attest to the fact that it's an old headline from from the Times, but cannot be precise about dates. There is also a New York Times from Aug 29, 1936 which references “an old famous London Times Weather Report”. Again, it doesn't have any information about the actual date it was used.

In The Times, November 3, 1939 there’s a letter to the editor

TO THE EDITOR OF THE TIMES

Sir,

-A little over a year ago I remember telling to an amusing little German in Berlin the venerable chestnut about our insularity which alleges that in the eighties of the last century there appeared the following heading in the columns of your newspaper: "Dense Fog in Channel: Continent Isolated for Three Days." My friend was much tickled, not so much by the story, which he had heard before, but because-as he explained to me-it is just the sort of story that is invented by an Englishman and told by Englishmen to amuse other Englishmen. Listening last night to the broadcast in English from Hamburg I heard this same chestnut rolling across the North Sea to us. With exquisite laboriousness the speaker explained and re-explained the point of the joke. The eternal German in him prompted him to add at the end: ' Why do you not laugh? For in Germany, you see, we have a sense of humour. We see the absurdity of a small island imagining itself so important that the Continent should be isolated from it." I wonder if my little German friend will ever meet the broadcaster from Hamburg, and if he will have the courage to explain to him the true point of the joke. I am, Sir, yours, &c., MORAY McLAREN. The Savile Citb, W.1, Nov. 1. A VENERABLE CHESTNUT

which complicates things by suggesting it was in use by the Nazi's in the 1940's. This letter also suggests it is from an old newspaper, but this time tries to be more precise by saying it was the 1880's. However, again, no headline with the title exists from that time.

Other alleged appearances from the period are a Punch cartoon from the 1940’s by Russell Brockbank (but I cannot find any of his cartoons with this quote in them), and this cartoon, by David Low from 1942, which perhaps helps to testifies to its use as a cultural meme rather than as a real newspaper headline.

Throughout the 1930’s and 1940’s it appears to have been appearing as a reference to a title from back in the 1880’s, and the current resurgence could be traced to the mid-1990’s when it appeared, firstly as a title in the Spectator,

http://archive.spectator.co.uk/article/27th-november-1993/8/fog-in-the-channel-continent-cut-off

and in a NY Times article which talks about the Euro-Tunnel.

r/https://www.nytimes.com/1994/05/07/world/england-and-france-now-a-train-trip.html

The article in the Spectator uses it without any attempt at suggesting the origins, and without the self-consciously mocking tone that later journalists used when it was connected to Brexit.

Interestingly, Sir Alastair Morton, then co-chairman of Eurotunnel, is said to have had a framed copy of a Times newspaper with this headline in his office during his period as chairman. He told his many guests that it was the front page of a copy of the Times from the 1930’s. This was testified to by many politicians and journalists during the 1990’s, when the tunnel was a constant source of news. Morton used this quote and idea as a way of endorsing the tunnel in the eyes of the influential people who came into his office, and it’s not difficult to suggest that this use allowed it to re-enter the political landscape where it became more prevalent in the following 20 years within discussions of the UK/EU relationship. If the actual newspaper in Morton’s office was “real”, then it could very well be a mock up that he used simply as a prop. The fact that no real headline exists advances the claim that he used a fake paper. Morton was interviewed by the Spectator several times in this period, and that Spectator article is the first time I can find it being used to refer to UK/EU relations.