Most natural is my nose is blocked or it is all stuffy. You use clogged when you have a blockage..in vein or sink pipe for example. You could use clogged up referring to the mass of snots in your nose, whereas blocked nose stops you from breathing. Stuffed refers to a filling in something, stuffed toys, stuffed chicken, I am stuffed = well fed…

All phrases imply that you have a cold. However, you would usually use "stuffed" in describing "feeling full" after eating e.g. "I feel stuffed after eating that large pizza!" Hope this helps :)