For a newspaper that has yet to put a great deal of marketing muscle behind its international presence, the New York Times isn't doing so badly. A third of its 59 million unique monthly users live outside the US, making it one of the most widely read news sites in the world – even before it really tries.

On Tuesday that effort to advance the paper's global reach begins in earnest with the launch of the International New York Times, an integrated digital and print venture designed to push the Times's revered journalism to a worldwide audience. Edited in Paris, Hong Kong and New York, it will build upon the foundations of the International Herald Tribune, the 126-year-old title that will cease to exist, ten years after the NYT bought out its partner for the previous 35 years, the Washington Post.

For Jill Abramson, executive editor of the New York Times, the rebranding is the culmination of a long-held ambition to create what she calls the "24-hour global newsroom". The new website and print edition will aim to reverse the recent flow of energy across the Atlantic that has seen UK news outlets – the Guardian and the Mail Online primarily – make inroads in the US. Now it's the turn of the Gray Lady to see whether she can turn the tide by taking on the famously cut-throat UK market.

Abramson thinks she can. "The thirst for the quality information the New York Times provides is very real and pretty unquenchable at this point," she says. But what can the NYT offer British readers that they don't already get? "Sophistication of analysis and news presentation that we do on a higher plane than anywhere else."

New York Times editors don't do modesty, Abramson no exception. She says her ambition for the paper is to become "the international provider of the highest quality news and information. That simple. It's the best, so everyone is going to want that."

The opportunity for growth – both in readership and income – is underlined by the statistics. Only a tenth of the NYT's paid digital subscriptions of about 700,000 come from outside the US, and though Britain is its third largest web audience the UK represents only 4% of total traffic (the US accounts for 66% and Canada 5%).

Britain has been on Abramson's mind a lot over the past few months. In August, she accepted an invitation from the Guardian to join in the exploration of the files leaked by Edward Snowden, a repeat of the 2010 collaboration between the two news organisations over WikiLeaks in which she was also heavily involved.

The latest joining of forces with the Guardian earned Abramson an approach from the UK embassy in Washington. In the NYT's first official comment on the incident, she says that an embassy figure whom she does not name asked to speak to her several weeks ago. "They were hopeful that we would relinquish any material that we might be reporting on, relating to Edward Snowden."

If that was an attempt to dissuade the NYT from publishing stories about GCHQ and the NSA, it didn't work. The British government had met its match in Abramson: "Needless to say I considered what they told me, and said no."

There's a calm confidence to the way she relates the story that is striking, and very American. She repeats one of her favourite expressions: "The First Amendment is first for a reason. It makes me feel a little like I'm pontificating to cite the founders of this country, but it's true they were so afraid of centralised power that they saw a free press as the critical bulwark against unbridled government – and that is our role."

While the First Amendment has largely shielded the NYT from the British-style threat of prior restraint, a new challenge to robust journalism has emerged in the form of President Obama's aggressive pursuit of official leakers. His administration has launched seven prosecutions under the Espionage Act, more than twice the number under all previous presidents combined.

"That's put a chill on sources, particularly those involved in national security matters," she says. "Very experienced investigative reporters in Washington have said to me the atmosphere for reporting has never been less hospitable, and that means that keeping the public informed has become more difficult. There are principled people inside the government who want to share certain information but are now too scared to talk because they fear prosecution."

It's not just central government that is putting pressure on news organisations by intimidating sources. Those same sources are also becoming increasingly challenging as the internet and the rise of the so-called "Fifth Estate" have given them a plethora of outlets to go to, from WikiLeaks to specialist blogs to even direct publication. Snowden himself made it clear that he decided not to bring his material to the New York Times because of the paper's decision back in 2005 to hold on to its story for a year into the Bush administration's warrantless wiretapping programme.

Abramson says she is unfazed by that, pointing out that the Times won a Pulitzer for the story that it did eventually publish Snowden may have seen it as a problem, but "at the time there were considerations that caused the story to be held for a while".

She sees no cause for existential angst in Snowden's conscious decision to go elsewhere with his leaked treasure trove. "I'm not surprised that sources have reasons for going to certain news organisations or not, I still think that many sources still prefer to come to the New York Times because we have global impact and the rest of the news media pay acute attention to the stories we publish. I haven't seen any diminishment in that."

In a recent New Yorker article on the Guardian, Abramson's predecessor as executive editor, Bill Keller, said that had he still been editing the New York Times, with its famed and impenetrable wall between news journalists and columnists, he would not have allowed Glenn Greenwald to have a byline on the story. In the same article, Abramson sidestepped the question of how she would have handled the role of Greenwald, a journalist and columnist for the Guardian who has led this paper's coverage of the NSA, saying she doesn't deal in hypotheticals.

This goes to the heart of one of the key questions now facing the New York Times in the melée of the digital revolution: can it hang on to its tried and tested rules when the world around it is changing so rapidly? Can regulations set in the days when newspapers were powers unto themselves still apply in the age of interaction and self-publishing, or is there a danger that the Times gets left behind because it is too rule-bound?

Again, Abramson is having no truck with this idea. "All I can speak to is the standards that the New York Times has for the journalists on the news side. We expect them to not express opinions publicly. The New York Times has taken the view that we see a reason to have a divide between news and opinion."

We are sitting in Abramson's office just off the famed New York Times newsroom floor. It is a theatrically cluttered space full of her varied knickknacks, including cushions embroidered with images of her beloved dogs, a DVD of a BBC docudrama on Richard Burton and Elizabeth Taylor and, quizzically, a book titled The Married Kama Sutra.

It's been two years since Abramson became the first female editor of the New York Times, and nearly six months since she was notoriously accused by unnamed sources in a Politico article of being "difficult to work with". The charge provoked a counterblast from the media analyst Emily Bell that argued such denigration was sexist.

So who was right, Politico or Bell? "I appreciated what Emily Bell wrote and I thought the Politico story was unfair," she says, weighing her words carefully. Is there any truth in the line that she is a tough manager? "I think it's not for me to say."

I ask her what she has been most proud of during the infancy of her editorship, and she cites the investigative reporting that has been done on global issues such as the rise of "princeling" families in China, Apple's labour practices and the textile business of Bangladesh.

She also lauds the "very vibrant multimedia form of reading" that is now a top priority for the organisation "where video and motion graphics and animation are embedded as an organic part of an article. This newsroom has been incredibly inventive in raising new trails."

That aspect of the job has brought her into close contact with the CEO of the New York Times company, former BBC director general Mark Thompson. There has been plenty of reporting in this area too, also based on anonymous sources, suggesting that relations between Abramson and Thompson are at best complicated.

In his search for a new economic model for the paper that would take it into a secure digital future, Thompson has been experimenting with innovations that appear to stray from his corporate bunker on the 16th floor of the Times building into the editorial realm. They include the creation of a team of video-makers that answers jointly to Abramson and to the company's business side.

Isn't that putting tanks on your lawn, I ask. The general manager of the video team does report jointly, she says, "but I don't think that's having tanks on the lawn". Who is in charge of multimedia? "I definitely am in charge of the multimedia effort." She says it in such a way as to brook no dissent.

I ask her how she felt about the New York magazine piece that reported Thompson saying (yes, another unidentified source) that he could be editor of the New York Times. "I have no idea whether he said that," she replies diplomatically.

And if he had said it, how would she feel then? "He can …, he has …" Abramson shakes her head from side to side and looks to me – in my purely subjective judgment – deeply pained. Then she says: "I don't know, I have no comment."

You get the impression that there's going to be more to say in future about that particular relationship. But for now the moment is Abramson's. The newsroom is buzzing, the Pulitzers are coming in, and as of tomorrow she has a whole new toy to play with that will take the formidable journalism of the New York Times to a global audience.