Well, until recently, they haven't had direct access to the leaks, so covering the Snowden story or reporting on what the Guardian said were all they could do.
However, ever since the British government forced the Guardian to destroy a laptop containing leaked information, and detained David Miranda and confiscated his belongings, the Guardian has decided to partner with the NY Times, as they believe that the First Amendment, and American jurisprudence surrounding it, still provides better protection of freedom of the press than Britain's laws, so sharing information with the NY Times helps ensure that the British government can't stop the reporting on this story.
However, ever since the British government forced the Guardian to destroy a laptop containing leaked information, and detained David Miranda and confiscated his belongings, the Guardian has decided to partner with the NY Times, as they believe that the First Amendment, and American jurisprudence surrounding it, still provides better protection of freedom of the press than Britain's laws, so sharing information with the NY Times helps ensure that the British government can't stop the reporting on this story.