A "very friendly and productive meeting" with British leader Boris Johnson kicked off Jacinda Ardern's busy schedule at the United Nations leaders' week. 

The Prime Minister held her first bilateral meeting with her UK counterpart, discussing the desire to move quickly to negotiating a post-Brexit Free Trade Agreement as soon as possible when the UK leaves the EU.

She also gave a speech to open the UN's climate action summit, telling global leaders that New Zealand is "determined to show that we can be the most sustainable food producers in the world".

"We are determined that New Zealand can and will play our part in the global effort", the prime minister said.

Global climate protest celebrity and Swedish schoolgirl Greta Thunberg had earlier almost burst into tears at the podium as she accused world leaders of "betrayal" and demanded: "how dare you pretend that this can be solved by business as usual".

A message from Pope Francis was also beamed in from Vatican. US President Donald Trump also made an unexpected stop at the high-profile global summit, even though aides signaled for weeks that he would not attend.

The earnest entreaties were the first salvos on a warm and breezy morning in New York as the chaos of UN leaders' week began apace.Roadblocks littered Midtown, as, black Chevy Suburbans – presumably not electric – with tinted window lined the streets, idling outside hotels and waiting to whisk leaders down to the sterile grounds of UN HQ on the banks of the East River.

• Donald Trump meeting could result in trade victory for Jacinda Ardern
• Excitement for Jacinda Ardern's bilateral meeting with Donald Trump
• PM: Trump's 'S...hole' comment 'hugely offensive'

Ardern's speech kicked off a particularly busy day for the Prime Minister – dubbed Mad Monday by her office – which winds up around 10pm (1.30pm NZT) after a full schedule of speeches and engagements.

In it, she said that to overcome the threat of climate change "we all must start with an honest appraisal of our current situation".

"Although New Zealand accounts only for 0.17 per cent of the global total our gross emissions have increased a little over 23 per cent since 1990 and our net emissions by 65 per cent," she said.

"But, also as with many here, things are starting to turn around. Our gross emissions peaked in 2006. Over 80 per cent of our electricity already comes from renewable hydro and wind.

"And we have begun an ambitious agenda."

At her meeting with Johnson, the prime minister raised the issue of Tariff Rate Quotas on New Zealand exports to the UK and reiterated New Zealand's position that our exporters should be no worse off post-Brexit. 

Jacinda Ardern and Boris Johnson spoke about security, post-Brexit trade, and the Rugby World Cup.

She pushed for assurances that skilled migrants from New Zealand maintained access to a post-Brexit UK. 

The leaders also discussed developments in the Gulf, including the attack on Saudi Arabia's oil facilities, engagement in the Pacific, and of course the Rugby World Cup. 

Her meeting was set to be followed by another with King Abdullah II of Jordan, and with US President Donald Trump. 

Trump has faced widespread condemnation from other world leaders for starting the process of pulling the US out of the Paris climate accord. The UN meeting is an attempt to hold the signatories of that deal to their commitments to reduce carbon emissions.

As well as the speech at the UN climate summit, Ardern will also give a headline speech at the UN Secretary General's climate action summit private sector forum.

Later in the day, the agenda will turn to the Christchurch Call, the Prime Minister's signature initiative to block the spread of extremism online. This will include meetings with Sheryl Sandberg, COO of Facebook, Susan Wojcicki, CEO of YouTube and Brad Smith, president of Microsoft.

It is expected that new announcements around the Christchurch Call, and next steps from the tech giants, will be announced some time during the day.

Her last event is a roundtable with the tech companies including Microsoft, Google/Youtube, Amazon, Twitter and Facebook – which will be represented by its new global government relations boss, former deputy prime minister of the UK, Nick Clegg.