Anscombe herself was a Catholic and recognized God as moral law-giver. For a while she was working on a concept called "Virtue Ethics," based on Natural Law via Aristotle. But it's hard, maybe impossible, for "Virtue Ethics" to answer the very first question that comes up: why be virtuous? Why NOT be a self-centered jerk? Why NOT be cruel?
I don't think anybody has devised any ethical system that effectively replaces God.
Of course there are ethical atheists and agnostics; but whether they are conscious of it or not, they are living off the (dwindling) capital of a post-Judeo-Christian civilization.
We’ve all sinned (an “ethical athiest”, or an ethical “christan”, buddist, etc) does not exist. Only Jesus is good-and only He can give His nature to someone to change them; they cannot become “ethical” on their own.