Jeremy Corbyn has again sought to pile pressure on the Conservative party over the status of the NHS in any future trade deal with the US, as Donald Trump arrived in London just over a week from polling day.

Despite the US president insisting that he would not be interested in access to the NHS if it was presented “on a silver platter”, the Labour leader said he remained deeply sceptical of Trump’s claims and vowed to tackle him personally over the issue.

“If that’s the case, why have these talks gone on for two years?” Corbyn said to the BBC. “Why have they been kept secret? Why was, on a freedom of information request, the document only produced in redacted form? And eventually the full text has been released.”

After Trump touched down for this week’s Nato summit late on Monday night, Johnson spent much of Tuesday trying to avoid the US president – conscious that he could derail his carefully controlled election campaign – instead travelling to Salisbury in the Tory election battle bus.

Quick guide What is Nato and why are its members meeting in London? Show Hide What is Nato? The North Atlantic Treaty Organisation (Nato) is a collective defence alliance between 29 North American and European countries. Founded in 1949, the treaty provides that if one country is attacked, all Nato members would collectively respond. Nato was set up to counter the threat of the Soviet Union. The 12 founding members were: Belgium, Canada, Denmark, France, Iceland, Italy, Luxembourg, Netherlands, Norway, Portugal, United Kingdom and the United States. Over the years the organisation has expanded to its current membership of 29. Bosnia and Herzegovina, Georgia, North Macedonia and Ukraine are recognised as states with aspirations to join. Why is it meeting in London? This week's summit marks a celebration of the 70th anniversary of the organisation. London was chosen partly because it was the location of the original headquarters of the organisation when it was founded. What is on the agenda? During the two-day gathering there will be addresses by the Nato secretary general, Jens Stoltenberg, a formal reception at Buckingham Palace hosted by the Queen, and a meeting of the North Atlantic Council attended by heads of state and government. The agenda features discussions about Russia, China and the future of arms control. There will also be a series of bilateral meetings between leaders - the most testing of which are likely to feature Turkey's Recep Tayyip Erdoğan looking for support for his country's recent incursion into Syria. What's the context? Infighting is a major issue. For the third summit in a row, Donald Trump is expected to renew demands that European allies and Canada step up defence spending. He is also unhappy with his French counterpart, Emmanuel Macron, over a tax on American tech giants including Google, Amazon and Facebook. For his part Macron has lamented Nato's "brain death" due to a lack of US leadership, and said the organisation needs a wake-up call. He insists that strategic questions must be addressed, like improving ties with Russia and how to handle an unpredictable ally like Turkey. In turn, Erdoğan has lashed out at Macron. Their very public argument bodes ill for the summit, which is being hosted by the British prime minister, Boris Johnson. Johnson will want who to smooth things over and downplay any links to Trump, who is unpopular in the UK.

Trump, who last time he was in London started a wave of rumours about NHS privatisation post-Brexit, appeared to have been carefully briefed to not upset Johnson’s election campaign. He insisted he had “no thoughts” about the 12 December election, while still noting that he supported Brexit and believed Johnson would “do a good job”.

Asked about discussions about access to the NHS, he said: “I don’t even know where that rumour started. We have absolutely nothing to do with it and we wouldn’t want to if you handed it to us on a silver platter. We want nothing to do with it.”

Corbyn was attending a reception with Trump at Buckingham Palace on Tuesday evening outside which noisy protests took place against potential risks to the NHS from a future US-UK trade deal.

Asked what he would tell the US president if they chatted, Corbyn said: “I will say, ‘Welcome to this country. I hope you’ll understand how precious our National Health Service is, and in any future trade relationship with the USA, none of our public services are on the table, none of our public services are for sale.”

For his part, when asked whether he could work with Corbyn if the Labour leader won the election, Trump said: “I can work with him. I’m a very easy man to work with.”

Conservative hopes that the US president would support their narrative on the NHS were buoyed when Trump rejected the idea that US healthcare firms could seek access. But any nerves about potential electoral damage from unscripted Trump interventions were stoked by the fact a planned brief exchange with the media at the US ambassadors’s residence in London spiralled into a freewheeling, 50-minute question-and-answer session.

Facebook Twitter Pinterest Donald Trump is in the UK for a Nato summit. Photograph: Reuters

Labour have sought to maintain pressure on the status of the NHS in future US trade talks, dismissing Trump’s insistence that he would not seek access to health markets ahead of a potentially crucial day in the election campaign.

Both Trump and Boris Johnson were scheduled to hold press conferences at the Nato summit, taking place in Watford, on Wednesday afternoon.

Johnson returned to London from Salisbury for talks at Downing Street with the German chancellor, Angela Merkel, French president Emmanuel Macron, and Recep Tayyip Erdoğan, the Turkish president, with No 10 saying they discussed Syria, Libya and terrorism.

Labour have repeatedly argued that post-Brexit talks with the US on a new trade deal could see the UK forced to accept American corporations gaining access to elements of the NHS, or pushing up drug prices. Last week Corbyn produced official documents citing US demands that the NHS would be “on the table” in such talks.

Johnson faces a potentially tricky day on Wednesday, not least stewarding an important Nato summit amid serious tensions among members about its future. Speaking to reporters, Trump condemned Macron for the French president’s depiction of Nato as “brain-dead”, calling this a “very, very nasty statement”.

At the same time, the prime minister will be aware of the volatility of a US president who routinely sabotages the news narratives of even hosts and allies, and is visibly unsettled by the ongoing impeachment proceedings against him, the subject of a series of late-night Trump tweets after his arrival.

In a speech to the summit, extracts of which were released in advance, Johnson will urge unity, saying that while Nato had endured 70 years, “peace cannot be taken for granted and even as we celebrate this anniversary we must ensure that our deeds match our words”.

He said: “Seventy years on, we are rock solid in our commitment to Nato and to the giant shield of solidarity that now protects 29 countries and nearly a billion people.”

In Salisbury, Johnson visited military veterans before touring a Christmas market in the city, facing both cheers and abuse from watching crowds. Speaking to reporters on the visit, the prime minister called Labour’s claims on the NHS “pure Loch Ness monster, Bermuda triangle stuff”.

Asked if Trump had assisted him with his comments, Johnson said: “I want to stress that under no circumstances whatever will any part of the NHS be for sale and this is periodically, metronomically produced in a completely fraudulent way by the Labour party in order to distract from the vacuity of their position on Brexit and their unwillingness to get Brexit done. From our point of view, it was never remotely credible and it was never on the table.”

As Trump faced protests over his visit on Tuesday night, Johnson said the need for good relations between the UK prime minister and US president was “just a geopolitical, geo-strategic fact”.

He added: “The UK’s role in Nato is to be the glue. We’re the second biggest player. We’re the country that historically has helped to bridge the Atlantic together and to bring Europe and America together, and that’s what we’re going to continue to do.”