English law does have duty of care obligations, though they are limited. It all depends on whether you owe a particular person a duty of care, by virtue of being a parent, a teacher, a policeman, etc. People who take up the position of care could be prosecuted on omission grounds. But for complete strangers? Hell no. Noone should have to put themselves out for other people. The line would be impossible to draw. There could be an argument for having a duty to help if your own life or health isn't put at any risk while carrying out that help i suppose, but then would that be an objective or subjective test? The perception of whether you are in danger when helping would differ from person to person. It wouldn't do to have laws in place which force people to act in this manner. One hopes that good nature alone is enough.
We only hear the odd silly story- like those morons who let that kid drown. But compare the amount of those types of stories you hear about, to the number of tedious rescue stories we read, and the picture we get is of a generally helpful and willing society- and the best aspect of this nature is that it isn't forced by law, and so we can be warmed in the knowledge that most people are nice out of inner compulsion.