This is a tough one! The boring answer is that my headcanon for the fate of mortals after death in Tolkien-verse is…. pretty much just whatever happens to mortals after death.
More specifically, I think the humans and other mortals of Middle Earth have a variety of beliefs about what happens to their spirits after death but no one knows for sure. (…Of course, the one MAJOR difference between Arda and the IRL is that human souls in Tolkien-verse can manifestly be detained from the afterlife, and otherwise bound to certain tasks or fates. But temporary postponements aside–)
I think the ambiguity of what happens after death, the mystery of it, is an important unknown; it’s the biggest question mark, the only question mark that matters to many characters and philosophies. Life and Death are exclusively the province of Eru, who remains silent; everyone else is fallible, childlike facing the universe, including the Valar. I think the fate of mortals being a mystery, and conversely, the source of life being a mystery, is invisibly at the heart of the Silmarillion and its dramas.
I do have a second answer that’s more story-specific, and that is that the Arda of the books is a trial-run of Earth, and that after the foretold Second Music rolls around, everyone who endured the first cycle will have a second run as mortals. They may not be the same specific people they once were, but rather the same collective energy and souls, variously rehoused. I’m trying very hard not to say “the lifestream”, but it’s like…. the lifestream. This is the Story Specific answer because I really really really need that human-ainur redemption arc. >_>
…And on a like SUPER META and personal level, I think human spirits are withdrawn back to the source of the Fire, where they cease to be isolated entities and return to a collective and holistic state of Being; not separate yet not alone, unburdened by whatever transpired during their lives. In truth, a gift– but only once one leaves behind the value of ‘selfhood’.
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