For someone concious you should ask permission, and if they aren't you are covered in most countries by implied consent. As long as you follow your training you shouldn't make matters worse.

Sir you have a nasty paper cut on that finger. Is it okay if I amputate your leg?

It is sad to say that you could train everyone in the UK but you'd still get a majority of people who wouldn't help others. Worse still, there are others taking advantage of the situation. Even those who are first aiders don't always do the right thing properly.