Boris Johnson has pledged to represent remainers and work with them as “friends and equals”, in a New Year’s Day message in which he sounded upbeat about the UK’s fortunes in the 2020s.

The prime minister, who is seeing in the new year on the private Caribbean island of Mustique, said he was “acutely aware” that millions of people did not vote for him despite the Tories’ landslide election victory.

In a statement full of familiar election slogans that he was “getting Brexit done” and the deal with the EU was “oven ready”, Johnson also thanked those who do not see themselves as natural Conservatives for backing him at the polls on 12 December.

He said: “I know that many of you do not consider yourself natural Tories and may only have lent me your vote. I am humbled by your support and will work every day to keep it. I am also acutely aware that there are millions of people who did not vote for me and were disappointed by the result.

“If you are one of them, I want to reassure you that I will be a prime minister for everyone, not just those who voted for me. I know that you love this country no less, simply because you voted for another party or wanted to remain.”

As part of his overtures to remain supporters he said he wished to work with them as “friends and equals”. He has spoken since his election victory of running a “people’s government”.

Johnson has so far surrounded himself with Vote Leave figures, including his top team of political and media advisers. His chief adviser and strategist, Dominic Cummings, director of communications, Lee Cain, and press secretary, Robert Oxley, all ran the successful campaign group to exit the EU in 2016.

The prime minister’s drift towards a hard Brexit was regularly criticised during the election campaign by Labour, especially over his insistence not to extend the transition period beyond 2020, which could lead to a no-deal scenario, and for not properly protecting environmental and workers’ rights.

Overcoming division in the country and moving forward was another theme of Johnson’s message, which was filmed to be shared on social media.

“That distinctive sound you may have heard at midnight as the bongs of Big Ben faded away was not the popping of champagne corks or the crackle of fireworks from your neighbour’s garden.

“Rather it was the starting gun being fired on what promises to be a fantastic year and a remarkable decade for our United Kingdom,” he said.

His “oven-ready” Brexit catchphrase was also redeployed. “That oven-ready deal I talked about so much during the election campaign has already had its plastic covering pierced and been placed in the microwave,” he said.

Beyond leaving the EU, he said one of the first bills to be passed through the Commons in 2020 would be legislation on NHS funding, that enshrines in law the injection of an extra £34bn a year.

Other domestic priorities include raising education standards and getting Britain back up the international school league tables.