Hi! I'm not sure really. But I think, that it was from the whole stress, the loss of a good friend, the case, emotions... There was a lot of speculations about this, but, I think, its because of the stress she had. Maybe someone else has another answer, but for me this would be logical.
I thought it may be from poisoned holy water that the priest threw at her friend but it hit her. Earlier in the montrer there was mention of possibility of sarin gas ou vx gas used to kill the priest in Spain and Emily asked the Italian priest if he used those to kill the victims during the excorcisms. He could have put some in the holy water. Runny nose is an effect from exposure to those toxins.
She was about to walk into a church doubting her 2 friends, but nose bleed proves the priest did have poison in his holy water that he was using ( some got on her) and reassured her they were good guys and that the priest was in fact a bad guy.
Emily's nose started bleeding at the end of the episode because while Emily was calming john down the unsub was still sprinkling the poison and some got one Emily while he was trying to arrests the unsub and from the other murders what triggered the nosebleeds were stress and Emily was stressed about the job and losing her best friend it was her emotions running high