Wilfred Frost’s memories of his father’s first summer party are patchy – but vivid. “I remember it being a very small affair at our house in Carlyle Square,” recalls the 33-year-old US TV presenter of the legendary annual bash that came to define London’s social calendar every July. “And I was told that the first guests to walk through the front door were Elton John and his wife, Renate.”

Wow. Things have moved on. Just two years later, the party itself was transplanted from Sir David and Lady Carina’s London home to the nearby communal gardens, where Wilfred recalls being “rolled out in absurd little matching outfits” alongside his brothers Miles and George to wait on the prime ministers, politicians, journalists, showbusiness stars, writers and socialites who continued to enjoy his father’s hospitality in ever greater numbers for the next 30 odd years.

The gap in London’s social calendar is as nothing to the one the journalist and TV giant left in both British and American cultural life when he died in 2013. And while the Frost family were forced to reckon with another tragedy two years later – when Wilfred’s older brother, Miles, died suddenly at 31, from an inherited heart condition of which he wasn’t aware – the rest of the world has continued to mourn Sir David’s absence from their screens.

“Aside that one moment every summer when there would be these banks of photographers outside our house, we never really thought of Dad as famous,” shrugs Wilfred. “Dad’s celebrity in our eyes extended to him being stopped at the airport and told what a great interview he’d done. And Dad would always pretend he didn’t like being stopped – but he bloody loved it.”

Would Sir David have agreed to ‘selfies’? “No,” chuckles Wilfred. “The digital age passed him by. I always laugh that Dad was of groundbreaking impact on the world of television… but couldn’t turn one on. We’d be in the pub when the phone would ring and we’d find ourselves saying the same thing every time: ‘No… press the button that says TV, then Sky…’

And our friends would say: ‘Who was that? A four-year-old?’ To which we’d reply: ‘No – Dad.’ Incidentally, one of the greatest TV presenters of all time.”

It has been of some comfort to the American viewers, who miss Sir David as much as we do, that Wilfred has inherited shades of his father’s chummy yet razor-sharp interviewing style, and brought a dash of what Vanity Fair magazine described as “vintage-BBC flair” to the CNBC business news show he has presented since 2014. With Closing Bell, one of the highest-rated programmes on the network, it doesn’t seem likely Wilfred can be enticed back to Britain quite yet. But the Oxford-educated former fund manager has spent the past five years digitising Sir David’s archives, with a view to bringing his father’s genius back to our screens one day.

“It’s still a little way off,” says Wilfred, who at 6ft 4in has the angular good looks of a 1930s line drawing and a pleasing lack of transatlantic twang, despite five years in New York. “But I have two projects in the pipeline that I hope will be a celebration of Dad’s work. And I’ve got to say that although, 10 hours in, the archive can get quite emotional,” he admits, eyes shining, “going through it all has been phenomenal.”

Having wrestled back control of The David Frost Show (1968-72) from the CBS archive, where the tapes had sat gathering dust for 50 years, Wilfred has uncovered gem after gem of celebrity footage. And, of course, he has the full 30 and a half hours of the Frost/Nixon tapes.

“Only six [hours] were broadcast originally, and only about two minutes are really out there now.” Because Sir David famously took on the networks, producing and syndicating Frost/Nixon himself, Wilfred assures me that claims of a huge profit made in the 2008 film were nonsense. “Dad always told me that until the movie came out 40 years later, he was still in the red with those tapes.”

Growing up playing football with Prince William and Harry (Diana was a close friend of Lady Carina’s and his younger brother George’s godmother), and having prime ministers and presidents to stay at the family’s Hampshire retreat, must have helped Wilfred keep the nerves at bay when interviewing everyone on his show from Tony Blair and Boris Johnson; John Kerry, Mike Pompeo, Usain Bolt and Alec Baldwin.

He laughs now to think of “Thatcher coming to lunch the Sunday after she was ousted – and us being more excited about the helicopter. And Bush Senior coming to stay, but being more starstruck by his Secret Service men.”

The moment I was most struck by Wilfred’s ability to stay strong under pressure was when he stood up in front of the hundreds of people assembled for one final summer party thrown in aid of the Miles Frost Fund in July 2016, and spoke movingly of the brother who went for his morning run and never came home. “We reinstated Dad’s party just once that year for Miles. And I think the memory of him, with Dad, was what made it so marvellous.”

Over the past three years, the Fund – which helps raise awareness towards improving diagnosis of hypertrophic cardiomyopathy or HCM, the heart condition of which Miles died – has raised £1.5 million and helped set up 17 testing centres across the UK. “Because HCM affects one in 500 people, which means that 120,000 people in the UK are carrying it, but don’t know,” explains Wilfred, instinctively fingering the Cartier bracelet that once belonged to Miles and now never comes off.

“And it’s not like all those people are on death watch. Although it wasn’t the cause of Dad’s death, it showed up in his post-mortem that he was carrying the gene – and yet it didn’t affect him at all. But if you know you have it, then you won’t go out sprint training one morning. And although it would have been a blow for Miles to know that he wasn’t able to stay fit in the way that he enjoyed, he could have lived.”

Wilfred tells me about a 13-year-old boy who tested positive for HCM at their centre in Belfast. “The parents were so grateful, but the boy could only see that he couldn’t be a centre midfielder anymore. So when he told us he was learning to be a goal-keeper instead, we arranged for David Seaman to go out there and give him a coaching session. It’s about adapting your life.”

But although the family like to hold on to the fund’s work as a way of making sense of Miles’s death, he says, “because then, you think, maybe it wasn’t really in vain…” He pauses. “It never outweighs the grief.”

As if the family hadn’t had enough heartbreak, at the start of last year Lady Carina was diagnosed with cancer. “It was advanced melanoma in her back, liver and lung,” says Wilfred. “And it would have been a case of ‘You’ve got six to 12 months and you can try chemo which might or might not work’ had it not been for the immunotherapy drugs she took, which, thank the Lord, have been transformational. So Mum has – touch wood – made a pretty full recovery. And she really is the most remarkable person. When I think about what she’s been through.”

The presence of that archive – and the idea of bringing some of her husband’s greatest moments back to our screens – has provided some solace to Lady Carina. And Wilfred tells me that he has continued to learn from his father as he comes across new material in the archives. “As much as I loved spending time with Dad as a kid, I only see now that so much of his value as an interviewer was not so much in the questions he asked as the answers he listened to.

“And when you watch Dad you realise there is no need to make TV interviews about yourself. In the political sphere now it’s either sycophantic or shouty and there seem to be so many agendas other than just getting the most interesting stuff out of a guest. ‘It’s about opening them up, not closing them down,’ dad would say. And I agree. Maybe that’s why I happen to be quite like him in terms of style.”

Would Sir David be horrified to see the state the country’s in now? Wilfred laughs. “People often ask me that: ‘Wouldn’t your dad be shocked to see how polarised we are?’ And I’m always surprised – because I think he would have thrived on it.”