I am not sure what you are asking here but you seem to be asking if the foam will cure on it's own if left for a longer time. The answer is no, it must be baked. If foam is left without baking for too long then the cell structure will collapse and you will just have a hollow skin of latex inside your mold. If you do not have an oven you can use to bake foam, there is a simple way to build an oven to do this. Build a box from urethane foam with an aluminum foil facing. This can be purchased at Home Depot and put together with duct tape. You can then use light bulbs for a heat source. This was much easier in the old days when you could buy 300 watt incandescent bulbs. There are probably lots of plans for this type of thing if you do a Google search.
Like he says - you do have to bake foam latex.
Liquid latex can be air dried without baking - you use it to form a thin skin, not a solid cast, but you can stick the skin onto a cushion foam build-up. Richard Swenssen is very good at using this method. He shoes how he does it in a series of videos: https://www.youtube.com/watch?v=hZw0CeO_w_8&list=PL36E8E1B91054...
Another option if you can't bake, think about using platinum cure silicone (Dragonskin, Platsil Gel-10, or similar) for your puppets instead. Silicone has its own advantages and disadvantages, but it does not need baking.