The itch is the result of your brain. First, we start with the suggestion that you not scratch your nose. That implies that your nose will itch. Sitting there, watching the clock keeps your attention focused on your nose. Any small sensation becomes the focus of your attention and quickly builds until you feel that you just have to scratch it.
Some people are more suggestible than others.
While some people may be able to resist scratching for the five minutes, most people will feel the itching sensation in much less than the time limit. As we have just seen, suggestion can be a powerful thing. It is used in advertising all the time. Just be glad that I picked an easy to scratch spot like your nose. Just imagine if I had told you to think about that spot between your shoulders that is so hard to reach.