How long does it take for symptoms of the mange to show on a cat?

after it has come in contact with another animal that had the mange

Mange is fairly unusual in cats. It takes about 12 to 14 days to show up. I would wash the cat in a shampoo such as Sebolux Shampoo as a preventative measure available at www.revivalanimal.com. Also, Revolution topical flea treatment treats and prevents mange mites as well. I believe that you still have to buy that from your vet.