"The first edition of Fantastic Beasts was commissioned back in 1918 by Mr. Augustus Worme of Obscurus Books, who was kind enough to ask me whether I would consider writing an authoritative compendium of magical creatures for his publishing house. I was then but a lowly Ministry of Magic employee and leapt at the chance both to augment my pitiful salary of two Sickles a week and to spend my holidays travelling the globe in search of new magical species."
-- Newt Scamander in the introduction to Fantastic Beasts and Where to Find Them
Eight years after the book was commissioned and Newt Scamander started travelling the world in search of magical creatures, Obscurus Books publishes the guide. Fifty-two editions of the book have now been published as of the late 1990s, and it has been an approved textbook at Hogwarts since its original publication.
In the introduction, Newt/Rowling writes "The rest is publishing history: Fantastic Beasts is now in its fifty-second edition." The Muggle version of the book was published in 2001, but in the timeline of the series this actually falls between books four and five, which places it in the mid-1990s on the timeline. Of course, the current edition would only have been published before that date, so we can't specifically say when the 52nd edition made its appearance. Logically, however, those fifty-two editions have to fit between 1927 and 1995, a span of around 68 years, so we might assume that a new edition came out at the very least every other year and, more often than not, annually. So the first edition is dated 1927 and the current edition would be dated approximately 1994.