They authorized it, you had the surgery. Not your problem. This is now squarely between your insurance and the hospital - not between the hospital, your insurance, and yourself. If either actually attempt to hold you liable, you just furnish your authorization notice (I wouldn't let whomever have the actual original however; a copy for those purposes should suffice). Nothing they can do. Not your fault, and certainly not your problem. Retain that authorization, though. It would be your only proof and protection, should they stupidly try and hold you accountable for THEIR backpedaling. And for that matter, the hospital, for not filing their claim in a timely manner in the first place (they ought to know better. I mean, I certainly wouldn't leave MONTH'S grace between services rendered and payment! That's begging to get stiffed, that is. Tsk. Tsk).
Local hospital tried for years to make us pay up for an ER visit (collections hounding and all). They had no legal standing, it was THEIR error, that they didn't attain authorization from insurance, and thereby received no payment (which they would have, if having done their job correctly). Not any fault of mine. (And just how they thought they'd actually be able to hold a 13 YEAR OLD financially liable, besides... well, LMFAO).
Don't worry about it. You are NOT liable for their mutual screw-up.