If you haven't already, take picures of the damage.
If there's gonna be any delay in someone correcting it, find out from your insurance co., if you can go ahead and use a shopvac, or rent a Rug Doctor type machine, and use it for the suction. Fans, de-humidifiers, and/or if you guys can go ahead and demo...remove the carpet yourselves, and still have it covered to get it replaced.
That's a lot of work, so if it's not too long of a wait, let the pro's take care of it. I feel for you. Mine got stuck in the Fill mode, and ran for two hours while I wasn't home. My husband was away, so his friend came over and cut up and remove an area rug, and got the fans etc.. started. He had brought a sump pump, but we didn't need that.
Mine did not have dry wall, but I'd watch for wetness wicking up your drywall or paneling, if it's a finished basement. That's where the wetness can get overlooked, and mold will grow in the walls.
Good Luck !! After seeing what Oprah had on recently. I had an after thought. Kids and families can get just as sick from harsh chemicals used in our homes. I'd be wary of what's used to clean up your mess, as it will probably take a lot, before you're all done. You don't want chemicals lingering either.