The normal musty smell from this kind of problem would usually take a number of days or even weeks to appear, because it is from fungal & microbial growth caused by the damp conditions. The growth of those organisms can't immediately spring up to a noticeable level immediately.
However there might already have been an existing problem, which adding moisture has exacerbated. Maybe this isn't the first time they've drowned their carpet? Maybe this time they had to "clean it harder" because of the effects of last time!
No point adding more water to the problem. If the underlay is properly soaked it really needs to be treated like a flood: carpet removing completely, reverse of carpet treated with biocide and drying, underlay ripped up and disposed of, subfloor treated with biocide and drying, underlay replaced, carpet professionally relaid and THEN cleaning.
The carpet itself could even be knackered, if it is tufted the latex adhesive may well be knackered, and a few weeks/months down the line the carpet will delaminate and go all puckered up.
Quite a bit more expensive than getting a pro in to do it properly in the first place!
Anthony forgive me because I don't know your level of skill/expertise in these things, but you have to be very wary about getting involved with problems like this. At present, this custome is thinking they can just call someone in who will wave some fairy dust around and everything will be better. Had one like this myself on Friday.
What they fail to appreciate is the problems water can cause when it gets to places it isn't supposed to be. Presuming the underlay is as soaked as it sounds, if you leave that carpet down and spray some smelly stuff on it, not only will you not solve the problem but you'll make it worse for the customer in the longer run.
Normally, this sort of damage is due to an escape of water whichj is covered by home contents insurance. However I fear this problem might not be covered because they were attempting a DIY carpet clean. I would advise them to get their insurance policy schedule out and read through the small print. That's as much input as I like to have in these situations!
Saying all that it could just be a particularly stinky wool carpet and they simply don't realise wool smells stronger when it's wet!