Hello,
I read carefully your query and understand your concern. 
Bloody sputum is also referred to as 
hemoptysis. 
Causes of bloody sputum include 
lung infection such pneumonia or 
bronchitis.
I suggest to do a chest xray to confirm the diagnosis. 
Meanwhile,I suggest using anti inflammatory medications such as 
Acetaminophen to relieve the inflammation. 
I recommend to use a Mucolytic to treat the cough.
I suggest to drink liquids and rest.
To treat the nosebleed you should use a saline 
nasal spray .
If the symptoms continue, I suggest using a cycle of antibiotics. 
Hope my answer was helpful.
If you have further queries feel free to contact me again.
Kind regards! 
Dr.Dorina Gurabardhi 
General &Family Physician