Boris Johnson will attempt to mark his election promise to “get Brexit done” by writing into law that the UK will leave the EU in 2020 and will not extend the transition period.

As MPs begin to be sworn in at Westminster on Tuesday, the prime minister’s team is working on amending the withdrawal agreement bill so that the transition, also known as the implementation period, must end on 31 December 2020 and there will be no request to the EU for a further extension.

A Downing Street source said: “Our manifesto made clear that we will not extend the implementation period and the new withdrawal agreement bill will legally prohibit government agreeing to any extension.”

Johnson’s move to make his manifesto promise legally binding was mooted during the election campaign. His pledge not to ask for another extension was used by the Brexit party leader, Nigel Farage, as the reason why he would stand down his candidates in 317 Tory-held seats.

The newly elected House of Commons is likely to have its first vote on Johnson’s Brexit plan on Friday, Downing Street said.

Late on Monday, the prime minister announced a small reshuffle of his cabinet, appointing Simon Hart as secretary of state for Wales after Alun Cairns stood down from the post over what he knew about an allegation that a former aide had sabotaged a rape trial.

The former culture secretary Nicky Morgan, who stood down as an MP, was re-appointed to the job, and simultaneously given a peerage. A replacement for the environment minister Zac Goldsmith, who lost his seat to the Liberal Democrats, is also set to be announced, with a much larger reshuffle expected in the New Year.

The Commons resumes on Tuesday with the timetable initially taken up with two days of swearing in MPs, followed by the state opening of parliament and the Queen’s speech.

Johnson’s spokesman said: “We plan to start the process before Christmas and will do so in the proper, constitutional way, in discussion with the Speaker.”

Under Johnson’s plan Britain is to leave the EU on 31 January once the bill passes, at which point the transition phase with the EU kicks in.

During this period, the UK is expected to leave the customs union and single market and enter new negotiated arrangements, but will follow most EU law like other member states. It will not have voting rights in the EU institutions.

There has been provision for the UK and EU to jointly agree, on a one-off basis, to extend that period by a further period of up to two years, under article 132 of the withdrawal agreement, which was hugely unpopular with with some Conservative MPs keen on a hard Brexit on World Trade Organization terms.

The governor of the Bank of England, Mark Carney, said Johnson’s election victory and majority in the Commons means the likelihood of the UK crashing out of the European Union has decreased.

“The worst-case scenario is effectively a no-deal, disorderly Brexit. The probability of that scenario has gone down because of the election result and the intention of the new government,” Carney said, speaking in a press conference after the publication of the central bank’s financial stability report.

“The scenario itself and the risks that we protect the system against has not itself changed, it’s just become less likely.”

The 109 new Tory MPs attended a private party with the prime minister last night to celebrate taking their seats, some in places that had been held by Labour for close to a century. They also posed for a photograph in parliament’s Westminster Hall.

Their first major task as MPs will be giving the withdrawal agreement bill its first reading, a formality not involving a debate, and then the second reading, which involves a debate as well as a vote.

Having both on the same day will need the formal approval of the new Commons Speaker, Lindsay Hoyle.

“We will present a bill which will ensure we get Brexit done before the end of January. It will reflect the agreement made with the EU on our withdrawal,” the No 10 spokesman said.

“The PM made clear during the general election campaign that he would be aiming for a Canada-style free trade agreement with no political alignment.”

While it is understood in Brussels that Johnson was unlikely to ask to extend the transition period, senior sources have suggested the EU itself could have sought to take the political pain out of such a move by asking for extra time.

Officials in Brussels have discussed such an option in recent days but the prime minister’s initiative would remove that safety net.

Such is the scale of the negotiation, the European commission president, Ursula von der Leyen, has already said that a number of key areas will have to be dealt with after 2020, including aviation and any special arrangements for the UK’s financial services sector.

A failure to agree on state aid and environmental standards, among others, would lead to a tough and likely protracted negotiation on the level of tariffs that would be applied to goods crossing the channel.

The block on an extension also piles on the pressure on the government to have the right infrastructure ready at ports through which goods move from Great Britain to Northern Ireland.