Depending on the size of the cob house you'd like to build and the amount of individuals able to assist you, your cob house can be constructed in as little as a week (for a team working on a large one room, or so) or months.
That's the thing about cob houses - you must mix every inch yourself (or with horses or tractors), and each loaf is applied individually. The mixing and building processes each take extensive amounts of time, which is why cob is not a good over-night building method.
However, it requires no power tools, and the materials can be combined with nothing more than your own hands and feet. Plus, you can build custom features into the house, like built-in shelves, benches, and other furniture, and add countless decorative elements.
The best way to know where or not you can legally build a cob house, let alone learn the amount of time it would take to construct, you should seek out professionals in sustainable, natural development.