The ceremony took place in Westminster Cathedral and was attended by only 30 guests, including the couple's son, Wilfred, born on April 29, 2020.
British royalty were among the 60,000 spectators at Wembley.
The birth is believed to be slightly premature but both mother and baby are said to be doing very well.
Sunak, who has been leading the UK government's economic response during the COVID-19 lockdown as the Chancellor of the Exchequer, will automatically step in as part of an established order of precedence in the UK -- which does not have a written Constitution.
Johnson spent the Monday night in the ICU at a London hospital in a move Downing Street said was a "precaution" should the British prime minister require ventilation to aid his recovery from COVID-19.
Announcing the news on social media, Symonds, 32, said the baby was named after the UK Prime Minister's grandfather Wilfred and her grandfather Lawrie, with Nicholas a nod to the National Health Service doctors - Dr Nick Price and Dr Nick Hart.
In an indirect reference to Labour's perceived anti-India stance over the issue of Kashmir, he noted: "There can be no place for racism or anti-India sentiment of any kind in this country."
Downing Street has not confirmed if Johnson had been allowed any visitors or seen Carrie Symonds, his heavily pregnant fiance, but said he had received many thousands of get well cards.
"The Prime Minister has been moved this evening [Thursday] from intensive care back to the ward, where he will receive close monitoring during the early phase of his recovery. He is in extremely good spirits," a Downing Street spokesperson said on Thursday.
Downing Street has said that the prime minister would not be returning to work immediately and will be based at the sprawling 1,000-acre estate in Buckinghamshire, south-east England, for a recovery period on medical advice.
They gave me a face mask so I got litres and litres of oxygen and for a long time I had that and the little nose jobbie, Boris said.
On the advice of his medical team, he will not be returning to work immediately, with UK Foreign Secretary Dominic Raab to continue deputising as he recuperates.
Johnson's hospitalisation was described as a "precautionary step" taken on the advice of his doctor
In a video message posted on social media, the British prime minister said that he will continue to lead the United Kingdom government's response to the deadly virus, which has claimed 578 lives.
An ardent backer of Brexit, Boris Johnson has a reputation for brashness, bombast, bending the truth, and really bad hair, which has earned him more than a few comparisons to United States President Donald Trump (who also happens to be a fan of his). Here are some interesting facts about BoJo, as he is called by the British media.
Voters gave Johnson the huge mandate so that he can get the UK out of the European Union by January -- no ifs, no buts.
Johnson, the former foreign secretary, secured 92,153 votes against 46,656 of his rival Jeremy Hunt in the battle for 10 Downing Street.
Johnson becomes the 14th Prime Minister to be appointed by Queen Elizabeth II, the 55th to hold the post of British PM and the third in fairly quick succession since Britain voted to leave the EU in June 2016 - following on from David Cameron and Theresa May.