That's how most fantasy writers who aren't called Tolkien work. The world starts out largely formless. As their protagonists move through it details become defined in a way that supports the narrative. If they continue to write in that world for long enough it ends up highly detailed. Doctor Who ran for six years and around 140 episodes before Time Lords where introduced.
To be fair to Tolkien, it's how fantasy writers who are called Tolkien work, too.
Yes, he had the Book of Lost Tales and Elvish philologies and mythologies to draw on, but he didn't INTEND to draw on those and really the Middle-earth we know and love from the four non-posthumously published volumes were created geographically to fit the needs of the adventure as the party traversed it.
Did he already have a range of Misty Mountains, a spider-infested Mirkwood, a Necromancer named Telvido/Thû/Sauron, and Elven King who lived undergorund in the middle of the forest who had beef with the dwarves over some jewels? All yes. But none of those examples were actually the same places or people (save for Sauron, who moved towers from one Mirkwood (Dorthonion), to the other (Lasgalen). He recycled characters and ideas from his Stable for the purpose of his bedtime story.
But the tale grew in the telling, and he found a way to graft on its imperfect geography to his earlier stories, by having Beleriand sink. If he had known he wanted to tie it to the Lost Tales from the get-go, he'd have made sure to mention that the Shire was in an obscure corner of Hithlum. While writing Lord of the Rings, as he realized he wanted to graft it onto Silmarillion, he originally envisioned Gollum has having been alive during the feats of Beren and Lúthien, and the whole matter with Morgoth and the Fall of Gondolin and the Voyage of Eärendil less than half a millenium ago. The huge timespans of the 2nd and 3rd Ages removing LR from the Great Tales emerged after principle writing on the book was complete, to work out the Appendices and iron out inconsistency problems.
If anything, the Lost Tales were his rough
Sketch of Mythology and language that he could draw on for an illusion of depth when writing The Hobbit and The Lord of the Rings. It only became truly extensively deep when he got down to reworking the Latter Silmarillion. And even then he never was able to finish his revisions on The Fall of Gondolin nor The Nauglafring/Nauglamír – those chapters of The Silmarillion were ghost-written by the author of
The Fionavar Tapestry fantasy novels, and CT generally considered this idea a mistake and that he never should have redacted and published a non-editorialized version of his father's incomplete campaign bible. Especially not in the wake of learning his father was still considering completely overhauling the entirety of the mythology to better match his understanding of the physical sciences of the real world (such as having Arda round from the get-go and the Sun and the Moon already in the sky because who could seriously believe they grew on trees that lit the world before then and giant mountain pharos lit the world before those?).
Tolkien was doing the same thing all engagers in secondary creation (as he put it) do – working it out and reworking it out and reworking it out as he went along.
The OP's concern about players wanting to bring in characters from their "stables" of PC rosters is in many ways a parallel one to Tolkien's issue he faced with grafting his published novels onto his unpublished
Mythology – the issue being that there may be a tonal divide between the two source materials because they weren't originally intended to fit together. Tolkien struggled and failed to update
The Hobbit in the '60s to fit the tone of LR and the Great Tales, because it tonally was never meant to be like the others, even if it always had the
Sketch of Mythology in the backpocket to use for throwaway line references. He got through the Trolls encounter with his re-write and abandoned it because doing so lost the magic and joy in the process.
But ultimately LR WAS tonally building on both
The Hobbit and the Great Tales, part of why many readers struggle with the tonal shifts in
Fellowship. And like how Tolkien's Tale 'grew in the telling', D&D campaigns can (and SHOULD, IMHO) adapt around the stories of the PCs, even if those PCs came from a different origin, because D&D is a collaborative medium. Even with a published setting, these are just guideposts for a world and narrative that the DM and the Party will forge together through the interplay of their characters and their choices.
All I can do is report my experience. I mean, you talk about a background being "due" a week before the campaign starts. Are they creating those backgrounds in collaboration with any of the players or only you? Are they creating those backgrounds in collaboration with you or are they presenting you with a fully formed background for you to approve? 
