When you bend copper, it becomes much more "springy" than its original state. The wire will almost certainly have been bent at some stage before you use it, so the first thing you need to do it get it back to its original "soft" state.
You do that by annealing the wire. Heat it with a gas torch till it is red hot, and allow it to cool in air.
After it is annealed, make sure you only bend it once, into its final shape. You can repeat the annealing and bending as many times as you like, but you can only make one bend that will "hold its shape" accurately each time you anneal the wire.
Leaving the wire wrapped around the cone for 24 hours (or even for a year) won't change anything.
Here are a some of videos of annealing copper and bending a spiral which might help. You could try bending a spiral, annealing the wire again, and then "expanding" the spiral to make the cone shape.
You might also get some ideas from websites and videos on bonsai - bent copper wire is used to "train" the tree branches into the desired shapes.