If you have a mustache it'll be a pain if your nose bleeds, right?
Excuse me. I just need a quick word with you. Your nose is bleeding.
I tried to rush onto the train, but the doors closed, squished my nose and made it bleed.
A nose with blood is a nose with a nosebleed. It is a bloody nose.
The readings are the kun'yomi vocab readings. You already know 鼻's reading from the kanji and vocab, and you should know the vocab reading of 血 as well. Put them together and you have はなち, though you want to change the ち to ぢ. Alternatively, it's possible (though less common, but it doesn't matter if you're turning it to kanji via an IME anyways) to write it as はなじ. I guess that blood added a little tenten (it represents the blood coming out of that nose!). Spurt spurt spurt.