The Chronicles of Narnia are fifty years old; they were published long before the current blockbuster era in entertainment. In fact, pretty much all of the great modern children’s fantasy (Susan Cooper’s “The Dark Is Rising” sequence, Lloyd Alexander’s “Chronicles of Prydain,” much of the best work of Diana Wynne Jones, Narnia, and so on) is at least twenty years old, and so was published in a very different world.
Once the blockbuster mentality came to the book world (in the US, this would be in the '80s with the rise of the mall bookstores and the increasing emphasis on the New York Times bestseller list), it was only a matter of time before some children’s book or series was a huge success, in the same way that other categories had breakout, massive successes – think John Grisham or Nicholas Sparks. It took children’s fiction a bit longer because that side of publishing had always concentrated on selling their books into libraries, which is a steady dependable business, rather than aiming for consumers, who were thought to be unlikely to have much money or control over how it was spent.
Of course, these days, we all know that tweens and young teenagers have more free disposable income than their parents, and that they’re much more likely to spend that money on “hot” items. So, when the right book came along, they jumped.
That’s all to answer the question of why no other book had done it first, but the OP was asking why this book. Timing certainly had something to do with it, but I suspect the real reason was that Harry Potter and the Philosopher’s Stone was a good-enough book with lots of elements that appealed very strongly to its target audience (not due to any calculation by Rowling, by all accounts, but just because that was the story she wanted to tell). So it became a huge word-of-mouth bestseller in the UK (there was essentially no promotion at first, and the first printing was tiny), and that enthusiasm infected the US and then spread around the world.
The fact that the books got better as they went along helped to keep the enthusiasm going, and Rowling’s real talents (and her books’ appeal to adults) added to the potential audience. She might not be quite as marvellous as her most zealous fans claim, but she’s a quite good writer with some impressive strengths.