In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
Nose does not break the wudu according to the majority of scholars.
In responding to your question, Sheikh Ahmad Kutty, a senior lecturer and an Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states:
The bleeding from the nose does not break the wudu according to the majority of scholars including the Prophet’s companions and the Maliki, Shafi`i, and the Hanbali schools.
However, the Hanafi School differs; they say that nose bleeding breaks wudu is broken.
You are free to choose the former view unless the bleeding is profuse, in which case, you should wash your nose and renew the wudu.
Having said this, I would also advise you to choose to wash the nostrils if it is hard for you. The rules are relaxed only because of hardship.
Almighty Allah knows best.