Arthur C. Clarke's short novel Dolphin Island was published in 1963. It's clearly set in a somewhat "future" world, probably quite near future, but can we narrow down its temporal setting any further? Is it set in the 20th century or the 21st? When can we start referring to it (or can we already) as being set in an alternate past, from our perspective, rather than in the future?
Early 21st century, probably the first half of the 2010s.
There's a few pieces of information scattered through the novel which allow us to pin down the approximate time period in which it must be set.
In Chapter 1, the opening paragraph of the story sets the scene in the then-distant 21st century:
Johnny Clinton was sleeping when the hovership raced down the valley, floating along the old turnpike on its cushion of air. The whistling roar in the night did not disturb him, for he had heard it almost all his life. To any boy of the twenty-first century, it was a sound of magic, telling of far-off countries and strange cargoes carried in the first ships that could travel with equal ease across land and sea.
In Chapter 3, we get a more specific fix:
Perhaps he might even drift to land, like those men who had ridden the Pacific currents on the famous Kon-Tiki, almost a century ago.
If 1947 is "almost a century ago", that sets an upper bound of 2047 on the setting of the story.
In Chapter 13, another specific year is mentioned:
Yet it was only 1881—not yet a century and a half ago.
Our upper bound is reduced to 2031. The same 1881 story of Mary Watson is referred to as "a hundred and thirty years ago" here in Chapter 13, and later in Chapter 19 as "a century and a half ago". Assuming that "a hundred and thirty" means between 125 and 135, we get a setting of between 2006 and 2016, more probably towards the upper end of this range.
In Chapter 3, the story is set relative to humanity's first ventures into space:
Above him the sky was ablaze with stars—the ancient ones of the old constellations, the brighter ones that had been put there by man in the fifty years since he had ventured into space.
This puts the setting at roughly 50 years since around the 1960s (first man-made satellite 1957, first man in space 1961, first man on the moon 1969).
A figure of fifty years is also mentioned in the Note from the Author at the end of the book:
The hovership described in the opening chapters does not yet exist, of course, but the first commercial "Hovercraft" (the VA-3 and the SRN-2) are now operating in Great Britain. In fifty years, such air-supported vehicles may well have grown to the size of the Santa Anna.
This would be written in 1962 or 1963 (the novel was published in '63, and the note mentions an article published in March '62).
As a side note, it seems that Dolphin Island is set in a future where world peace was achieved some time around the 1970s. From Chapter 17:
"You may be quite right," answered the Professor. "That's what I want to find out. Even if the job's possible at all, it may not be worth doing. And even if it's worth doing, it may take several lifetimes. But one has to be an optimist; don't you remember the history of the twentieth century?"
"Which bit of it?" asked Steve. "There was rather a lot."
"The only bit that really matters. Fifty years ago, a great many people refused to believe that all the human nations could live in peace. Well, we know that they were wrong; if they'd been right, you and I wouldn't be here. So don't be too pessimistic about this project."
Suddenly, Steve burst into laughter.
"Now what's so funny?" asked the Professor.
"I was just thinking," said Steve, "that it's been thirty years since they had an excuse for awarding the Nobel Peace Prize. If this plan of yours comes off, you'll be in the running."
"Fifty years ago" would be around the time the novel was written, the early 1960s, when nobody believed world peace was possible. But it seems that it was achieved before the early 1980s, "thirty years" ago, since there was no need for Nobel Peace Prizes since then.