Fantasy fiction as a distinct genre involved in different markets based on a variety of influences - it's not a case of any one particular nationality having an oversized influence on the market as much as it is certain key works get more popular attention.

Around the turn of the century there were several more-or-less simultaneous developments in fantasy in various nations. In the United Kingdom, you had writers and works like Arthur Machen's The Great God Pan (1894), William Morris' The Well at the World's End (1896), M. R. James' Ghost Stories of an Antiquary (1904), and Lord Dunsany's The Gods of Pegāna (1905); in the United States writers like Robert W. Chambers' The King in Yellow (1895), L. Frank Baum's The Wonderful Wizard of Oz (1900), and Edgar Rice Burroughs' A Princess of Mars and Tarzan of the Apes (1912); in France you had Guy de Maupassant's "Le Horla" & "L'Homme de Mars" (1887), many scientific romances by Jules Verne, Gaston Leroux' Le Fantome de l'opera (1910); in German literature Franz Kafka, Gustave Meyrink - especially Der Golem (1910), and Hanns Heinz Ewers, etc. - and these were just a handful of the most prominent writers that were working in fantasy or fantasy-adjacent fiction. Some found book publication, others had more success with magazines - in the United States, especially, the 1910s saw the beginning of pulp magazines, which became the main market for a very American breed of fantasy fiction, spearheaded by writers like H. P. Lovecraft, Robert E. Howard, and Clark Ashton Smith in the 1920s and 30s.

But none of these writers were creating in a bubble, so to speak. English works by Dunsany, Machen, William Hope Hodgson, etc. were being consumed by American audiences, and are cited in the fiction and letters of Lovecraft & other American writers, and to a degree this is true on the reverse - American stories and pulps were repackaged and published for UK, Canadian, Australian, etc. audiences. The American pulp Weird Tales, for example, had both British and Canadian versions during the 1930s and 40s, and many American fantasy stories from Weird Tales were packaged in the Not at Night anthologies and sold in Great Britain - there was an interruption in this kind of trade during the World Wars, due to embargoes and paper shortages, but it picked up again immediately thereafter.

So too, translations of fantasy and related works from English to other languages and other languages into English were fairly commonplace. If we look at H. P. Lovecraft's seminal essay Supernatural Horror in Literature (1927 version) for example, devotes Chapter VI to "Spectral Literature on the Continent" and wrote:

On the Continent literary horror fared well. The celebrated short tales and novels of Ernst Theodor Wilhelm Hoffmann (1776–1822) are a byword for mellowness of background and maturity of form, though they incline to levity and extravagance, and lack the exalted moments of stark, breathless terror which a less sophisticated writer might have achieved. Generally they convey the grotesque rather than the terrible. Most artistic of all the Continental weird tales is the German classic Undine (1811), by Friedrich Heinrich Karl, Baron de la Motte Fouqué. [...] But France as well as Germany has been active in the realm of weirdness. Victor Hugo, in such tales as Hans of Iceland, and Balzac, in The Wild Ass’s Skin, Séraphîta, and Louis Lambert, both employ supernaturalism to a greater or less extent; though generally only as a means to some more human end, and without the sincere and daemonic intensity which characterises the born artist in shadows. It is in Théophile Gautier that we first seem to find an authentic French sense of the unreal world, and here there appears a spectral mastery which, though not continuously used, is recognisable at once as something alike genuine and profound. Short tales like “Avatar”, “The Foot of the Mummy”, and “Clarimonde” display glimpses of forbidden visits that allure, tantalise, and sometimes horrify; whilst the Egyptian visions evoked in “One of Cleopatra’s Nights” are of the keenest and most expressive potency.

So the germination and development of fantasy literature was really an international affair, with many different influences contributing to the development of fantasy in different markets. However, not all writers receive the lasting fame and impact on literature as others. The impact of Robert E. Howard's Conan the Cimmerian stories and H. P. Lovecraft's Cthulhu Mythos is not to be understated as an influence on fantasy, but their impact was largely limited before the boom in paperback fiction in the 1960s and 70s brought them to print in a wider market. J. R. R. Tolkien and C. S. Lewis scored success with The Hobbit (1937) and The Lord of the Rings (1954-1955), and The Chronicles of Narnia(1950-1956) respectively, which overshadowed other important but less widely-read authors like American author Poul Anderson, whose novels Three Hearts and Three Lions (1953) and The Broken Sword (1954) proved so influential on designer of Dungeons & Dragons Gary Gygax and British writer Michael Moorcock, whose Elric series owes much more to Anderson than Tolkien. Gygax' own "Appendix N" in the Advanced Dungeons & Dragons Dungeon Master's Guide (1979) gives a good idea of the kind of influences at work on what we today consider contemporary fantasy:

Inspiration for all the fantasy work I have done stems directly from the love my father showed when I was a tad, for he spent many hours telling me stories he made up as he went along, tales of cloaked old men who could grant wishes, of magic rings and enchanted swords, or wicked sorcerors and dauntless swordsmen. Then too, countless hundreds of comic books went down, and the long-gone EC ones certainly had their effect. Science fiction, fantasy, and horror movies were a big influence. In fact, all of us tend to get ample helpings of fantasy when we are very young from fairy tales such as those written by the Brothers Grimm and Andrew Lang. This often leads to reading books of mythology, paging through bestiaries, and consultation of compilations of the myths of various lands and peoples. Upon such a base I built my interest in fantasy, being an avid reader of all science fiction and fantasy literature since 1950. The following authors were of particular inspiration to me. In some cases I cite specific works, in others, I simply recommend all of their fantasy writing to you. From such sources, as well as any other imaginative writing or screenplay, you will be able to pluck kernels from which will grow the fruits of exciting campaigns. Good reading!

• Anderson, Poul: THREE HEARTS AND THREE LIONS; THE HIGH CRUSADE; THE BROKEN SWORD

• Bellairs, John: THE FACE IN THE FROST

• Brackett, Leigh

• Brown, Fredric

• Burroughs, Edgar Rice: “Pellucidar” series; Mars series; Venus series

• Carter, Lin: “World’s End” series

• de Camp, L. Sprague: LEST DARKNESS FALL; THE FALLIBLE FIEND; et al

• de Camp & Pratt: “Harold Shea” series; THE CARNELIAN CUBE

• Derleth, August

• Dunsany, Lord

• Farmer, P. J.: “The World of the Tiers” series; et al

• Fox, Gardner: “Kothar” series; “Kyrik” series; et al

• Howard, R. E.: “Conan” series

• Lanier, Sterling: HIERO’S JOURNEY

• Leiber, Fritz: “Fafhrd & Gray Mouser” series; et al

• Lovecraft, H. P.

• Merritt, A.: CREEP, SHADOW, CREEP; MOON POOL; DWELLERS IN THE MIRAGE; et al

• Moorcock, Michael: STORMBRINGER; STEALER OF SOULS; “Hawkmoon” series (esp. the first three books)

• Norton, Andre

• Offutt, Andrew J.: editor of SWORDS AGAINST DARKNESS III

• Pratt, Fletcher: BLUE STAR; et al

• Saberhagen, Fred: CHANGELING EARTH; et al

• St. Clair, Margaret: THE SHADOW PEOPLE; SIGN OF THE LABRYS

• Tolkien, J. R. R.: THE HOBBIT; “Ring trilogy”

• Vance, Jack: THE EYES OF THE OVERWORLD; THE DYING EARTH; et al

• Weinbaum, Stanley

• Wellman, Manley Wade

• Williamson, Jack

• Zelazny, Roger: JACK OF SHADOWS; “Amber” series; et al

The most immediate influences upon AD&D were probably de Camp & Pratt, R. E. Howard, Fritz Leiber, Jack Vance, H. P. Lovecraft, and A. Merritt; but all of the above authors, as well as many not listed, certainly helped to shape the form of the game. For this reason, and for the hours of reading enjoyment, I heartily recommend the works of these fine authors to you.

– E. Gary Gygax, AD&D Dungeon Masters Guide (1979) 224

Gygax' list is light on non-English-language sources, but there is a fair amount of non-English writers influencing the above.