This trend began within a few years of publication, at least as early as 1961
The first mention I can find of this theory is in a 1961 issue of the fanzine Sex in Science Fiction, in an article by Marion Zimmer Bradley titled "How Dragged Were My Heels".
Bradley doesn't seem to agree that the Frodo and Sam relationship should be classified as homosexual, but she definitely floats the idea.
And some writers --Judith Merril in a book whose name I have forgotten--glanced on the possibility that future spaceship crews or colonies on the moon, all-male, would have to compensate for possible repressed homosexual desires in themselves. Jerry Bixby, in SHARE ALIKE, has made almost the same point in a rather shuddery story of a man who enjoys contact with a (male) vampire. But must we now have a rash of stories whose authors feel conscientiously obligated to deal with the inevitable problem of homosexuality 'rearing its ugly head' in every spaceship story from now on, just because that problem is well and truly dealt with in one or two such stories? The writing on the wall might indicate that perhaps we must. The context of today decries male friendship to the point where the Hero and his Faithful Sidekick are no longer an acceptable story-material.....someday remind me to print my commentary on the something-or-other person who described Batman and Robin in those terms. And yet I wonder if this can't be as grotesque as the compulsive attempt to prove via fiction that this hero and his girl couldn't live chastely in a spaceship for two weeks. I'm thinking of Tolkien again, perhaps because this point recurred so often in my paper on Men, Halflings and Hero Worship. The "Mount Doom" sequence, Sam's care for and emotional preoccupation with Frodo, the growing closeness and intimacy between them, the intensely emotional and moving quality of their love for one another, certainly places this in the category of classic David-and-Jonathan friendships. And you know what our hypothetical compulsive seeker of Freudian realism would say about that!
I emphasize this point because it was forcibly called to my attention when a friend suggested that I should include the last two of the Tolkien series on my annual listing of fiction dealing with homosexual and variant themes. I pooh-poohed the notion, as did my co-editor. However, once the point had been raised in my hearing, I discussed it -- not to put too fine a point on the matter --with several persons who, to my knowledge, were capable of at least an emotional response and/or reaction to their own sex. All agreed on one point; that this portrayal was both innocent and at the same time particularly moving for the quality of unselfish love and devotion expressed. All -- to my mild surprise, being used to hearing some of these biased critics demand that David and Jonathan must be openly claimed among the homosexual aristocracy -- were in absolute agreement that Tolkien's "real" meaning could safely be left to a psychoanalyst and that the books should NOT be included on such a listing!
Sex in in Science Fiction (1961) - Digitized as part of Marquette University's Tolkien fanzine digitization project.
The Return of the King was published in 1955, making this just six years after publication. It is very possible that there are earlier fanzines or other publications that have suggested this, but if so they seem to have been lost to time. Note though that Bradley talks about this as if it isn't a novel idea.
So I think it's safe to assume that merited or unmerited, this idea was being discussed since the very beginning.