Tolkien directly influenced DnD which shaped a lot of fantasy to follow. Final Fantasy, f'rex, lifted a ton of stuff directly from DnD (Hi Saguagin Sahagin!). That will shape the look of a lot of fantasy to come.
It's probable a different fantasy would influence Gygax and produce a DnD with a different flavor. However he also only adapted Tolkien, he threw everything and the kitchen sink in so plenty of stuff will still make it. DnD's most notable takes from Tolkien are (IMO) races (Pre-Tolkien elves were seen as tiny gnome-like creatures that were crafters as in the Shoemaker and the Elves, not tall skinny ubermensch). Secondarily the party (Fellowship) format and magic items were lifted there.
DnD inspired by another source will probably have a different selection of races, Halfings, Elves, Dwarves, and Orcs are all lifted directly from Tolkien. If Gygax pulled on Barsoom and Conan for his intial inspiration we might see humans, green martians with the serial numbers filed off, trolls, jaguar/beast men, and perhaps something like lizardmen lifted from some sci-fi story. Magic items will be less prominent, Conan has a dependable steel sword rather than a magic one. Possibly we'd see materials (Stone<Bronze<Iron<Steel<Adamantite<Orichalcum) being the standard for better gear rather than enchantment. Mithril was all Tolkien so it will never show up. Magic jewelry and weapons don't show as nearly as pivotal in works prior to Tolkien.