This is my best guess, though it's not the only book of its kind:
Tomorrow Revealed, a novel by John Atkins; originally published 1955 in Britain, reprinted 1956 in the U.S. and Canada, all editions in hard covers. (P.S. The original poster points out that the book is available at the Internet Archive.)
From a review by P Schuyler Miller in Astounding Science Fiction, June 1957, available at the Internet Archive:
This is a pure tour de force whose price and special nature will probably mean that you buy it off a remainder table in a few months or a year. Since there is no plot and no TV or film potential, there is not likely to be a paperback edition (Roy seems to have imported the printed pages of the English book).
The idea is one I used years ago in a guest editorial for—I think—Startling. It was called "Counterfeiting a Golden Age," and suggested the perplexity of future historians who try to reconstruct the history of our times from a file of science-fiction magazines. And the narrator of "Tomorrow Revealed," who has blundered on a library somewhere along the upper Nile in 3750 A.D., has set himself exactly that job: reconstructing some eighteen hundred years of the Dark Ages of his own time from the books of H. G. Wells and George Orwell, Ray Bradbury and A. E. van Vogt, C. S. Lewis and Wilson Tucker, Aldous Huxley and Vargo Statten.
The sources he has used are all English—Bradbury's "Martian Chronicles" is cited by its British title, "The Silver Locusts," and one of the key references, Robert Graves' "Let [sic] the Northwind Rise," has its original title, "Seven Days in New Crete"—and a few may be unfamiliar. But by and large, most reasonably well-read science-fiction devotees should be able to appreciate the ingenuity with which the author has woven together and reconciled the hosts of conflicting elements in his "sources," to produce a consecutive, consistent "history" of the next two thousand years.
Most of H. G. Wells' accounts, for example, are badly out of phase with the "histories" written half a century after his time: this is explained by pointing out that Wells lived in an era—that of "1984"—when it would have been politic to back-date the Martian invasion and Cavor's visit to the Moon, and that other details of both chronicles were adjusted to satisfy the temper of the times. The Vitons of Eric Frank Russell's "Sinister Barrier" become Earth's wicked eldils, described by C. S. Lewis in his "Out of the Silent Planet" and "Perelandra." And so it goes. I have counted some fifty irreconcilable books woven into this amazing history of the future, and other authors—Kuttner, del Rey, Blish, Kornbluth—are mentioned when their short stories contribute to the framework.
The closest parallel is, of course, Stapledon's "Last and First Men"—which is so nonconforming to the rest of the canon that it is passed off as "fiction." The style is matter-of-factly narrative, as it should be. And, in the doing, Mr. Atkins has shown how extensively the minds of science-fiction writers have traveled along the same paths in their projections of Man and his society. SF may not be a literature of prophecy, but it has written its own formula for the future, which John Atkins has translated back into words.
Alternatively, here is a more obscure work in the same vein and from about the same time:
A Short History of the Future: Based on the Most Reliable Authorities With Maps, a 1955 hardcover novel by R. C. Churchill. Probably not the one you're looking for, since you didn't mention any maps. Here is a review from NESFA's Recursive Science Fiction site:
It is not exactly fiction, in that it assumes that science fiction is actually predictive and tries to reconcile major works into a coherent timeline. Oppressive societies described by Bradbury, Vonnegut and Orwell coexist on Earth while Bradbury’s The Martian Chronicles occurs on Mars. Then, Bradbury’s nuclear war leads to Huxley’s Ape And Essence and to works by other authors. Apparent inconsistencies are explained humorously so that implausibilities do not really matter.