It will cure for a long long (even years for slight residual curing)time. A lot depends upon the mix and how thick it was. It is not uncommon for concrete drives to take a month or more to cure out to its full compression strength if it is cold, etc. But usually that means you don't want heavy vehicles driving on it or it will break. They do compression tests on it over time till it reaches its design compressive strength which may be 3000 lb., 4000 #, etc.
The curing/drying process is actually the portland cement absorbing water/moisture in a chemical process.(basically a moisture exchange process) That is why in hot/dry weather you have to sprinkle/spray water on the surface every day or else the surface will cure out quicker than the middle and then start to crack and crumble away. You also have to control the moisture on the ground before pouring concrete such as wetting down the sand (or even let wet sand dry out a little) or else you get a different cure rate at the bottom of the concrete which also causes problems. You want an even cure rate throughout. A lot goes into the process of new 12" thick concrete on our interstate highways. It's not rocket science but there are many pouring concrete who don't really understand what they are doing or the entire process. See if the contractor can tell you off the top of his head what the slump factor was, the mix amount of water, bags of cement, size of aggregate, compression strength, and if what additives are in the concrete. Then see if he passes the test.
I'm not sure curing is your problem now after a month. Even cured concrete will allow moisture up through it which is why we have wet basements. Did he place any plastic between the ground and the concrete he poured. Was this poured on or after a rainy day or freezing weather or on freezing ground?
Are you talking about a sump basin for your french drain or is this one of those french drains installed around the interior walls of the basement (instead of outside)? Also how far away from the french drain cavity is the concrete still wet? Please describe what you had done to your basement and what problems you had that caused you to get this french drain system. I can't picture it from here.
If this is an interior french drain cavity around the interior walls, then you probably don't want to build and carpet up to the edge anyway cause it will need some room. I wouldn't worry too much about a treated lumber bottom plate exposed to damp concrete though.
I like carpet in a dry basement and others around here don't. Just remember you need to get the floor moisture under the carpet under control. If not then you start having mold, mildew, and early carpet replacement problems. Mold and mildew will start thriving in 70%+ humidity. What is your basement humidity?