In English, adjectives generally follow a specific order: quantity, opinion, size, age, shape, color, origin, material, and purpose. Based on this order, here are the correct arrangements for your examples:

A big \ fat \ ugly \ white cat. A nice \ huge \ red \ Italian umbrella.

To clarify:

In the first example, "big" and "fat" are size adjectives, and "ugly" is an opinion adjective, while "white" is a color. In the second example, "nice" is an opinion adjective, "huge" is size, "red" is color, and "Italian" is origin.