Dawkins is a liberal and an atheist. He can criticize the Anglicans as an atheist and the Catholics as both a liberal and an atheist.

Liberalism doesn't come from atheism per se, since atheism per se is just the absence of belief. However, liberalism can come from the positive philosophies many atheists do hold, such as materialism, utilitarianism, and Epicureanism. Or from the fact that the atheist lacks religious dogma telling them to support conservative sexual values.