The Met told me in response to a recent Freedom of Information request that the investigation is "inactive pending further information being received." Since 2014, I've had several updates from the Met regarding the investigation, and this marks the first time that its status has changed from "ongoing." In November 2017, the Met stated that it was a "complex investigation and enquiries continue."
The investigation, which was given the code-name Operation Curable, had been led by the Met's Counter-Terrorism Command, under the direction of assistant commissioner Mark Rowley. In March 2018, Rowley retired from the Met -- and with his departure, it seems the Curable investigation went cold.
The majority of the documents in Snowden's leaked archive revealed classified American mass surveillance operations. But a significant portion of the files disclosed explosive information about electronic spying programs operated by the UK’s largest intelligence agency, Government Communications Headquarters, or GCHQ.
British authorities responded furiously to the Snowden revelations and tried to prevent The Guardian from publishing them. Infamously, representatives from GCHQ were sent to the newspaper's London offices at one stage to oversee the destruction of hard drives that contained the secret files.
The police went as far as to argue that publishing the Snowden files was itself a terrorist act, thereby explicitly conflating journalism with terrorism. In August 2013, a memo authored by the Met and domestic spy agency MI5 asserted that “the disclosure [of the Snowden documents], or threat of disclosure, is designed to influence a government and is made for the purpose of promoting a political or ideological cause. This therefore falls within the definition of terrorism.”
In December 2013, one of the London force’s most senior officers, Cressida Dick, was questioned about the case during a parliamentary hearing. She acknowledged that the force’s investigation was looking at whether reporters at The Guardian had committed criminal offences -- some carrying potential 10-year prison sentences -- for their role in revealing secret surveillance operations exposed in the documents. “We need to establish whether they have or haven’t [committed offences],” Dick said. “That involves a huge amount of scoping of material.”
It is unclear how much taxpayer money and police resources were invested in pursuing the Curable investigation. The Met has declined to provide any information about the amount of funds spent on the probe, or disclose the number of officers who worked on it; the force claims that it does not hold records of these details. It is also unclear whether the investigation may at some point resume. The Met said that the probe is inactive pending further information being received -- what that information may be, and whether it will ever actually materialise, is anyone's guess.