In Australia you must get consent before applying first aid, unless they are unconscious and cannot provide consent. This leads to an interesting potential situation where a bleeding person could refuse consent, and you have to wait until they pass out before attempting to apply first aid. If you apply first aid against consent, you can be charged with assault, so in the training they teach you to always ask first.
The Australian first aid law also talks about "duty of care" - if you are trained in first aid you have a duty of care in basically any situation in which you have a relationship with the person requiring aid. For example, if a friend, family member, or work colleague required first aid and I refused, I could be charged with negligence. If I was walking down the street and saw someone have a heart attack, or saw a horrific car crash, I would be under no legal obligation to help (moral obligation notwithstanding). However, if I were to talk to the victim, e.g. ask if they are ok or need help, then a relationship has been established and I would be legally obligated to apply first aid until a trained responder arrived.