One of the body's responses to cold temperatures is to constrict blood flow to non-essential areas and redirect it toward the heart to preserve body warmth. This response is the main reason why your nose turns red when it's cold.
The redirection of blood flow in cold temperatures is an instinctive survival response. The blood vessels in the extremities constrict, reducing the flow to the extremities -- including the nose -- and leaving more blood for your vital internal organs.
As a result of this limited flow of blood, the nose and other extremities will initially go pale. The body will periodically dilate the constricted vessels to allow a brief burst of blood to flow into the area and then constrict the vessels again. This burst of blood causes the nose to turn red.
A sudden increase in your surrounding temperature, such as when you go inside from being out in cold weather, will cause the constricted vessels to quickly re-dilate. The increase in blood flow will cause your nose to turn red until the blood flow equalises and the skin returns to its normal colour.