Boris Johnson has announced plans to project a giant clock face on to Downing Street on the evening of 31 January, in a bid to move on after a backlash over his failure to get Big Ben to bong for Brexit.

Diehard Brexiters, including Conservative MP Mark Francois and Brexit party MEP Richard Tice, had reacted furiously to news that Big Ben would not ring out to mark the UK officially quitting the EU at the end of this month.

Francois said earlier in the week that it was “inconceivable” that “the most iconic timepiece on Earth, which is Big Ben”, should not be used to mark Brexit.

Asked about the controversy on Tuesday, the prime minister said the government was “working up a plan so people can bung a bob for a Big Ben bong”.

Several crowdfunders were set up to help raise money, one of which had amassed more than £227,000 by Friday afternoon. But in an increasingly heated and farcical atmosphere, it subsequently transpired that the government had no such plan, and the House of Commons commission had no way of accepting public donations.

In a bid to draw a line under the row, No 10 has now arranged a light show, featuring a clock counting down the minutes until 11pm – midnight Brussels time – when Britain will formally leave the European Union.

Union flags will fly in parliament square, the scene of lively pro- and anti-Brexit protests over many months; and a commemorative coin will finally go into circulation – a previous iteration having been melted down, when Brexit was postponed.

Government buildings on Whitehall will also be lit up for the occasion, and Johnson will deliver a televised address to mark the moment. The costs of the celebration will be borne by the taxpayer.

Despite Johnson having won the Tory leadership with the backing of hardline Brexiters and expunging many moderates from his party, he now insists he is keen to heal the divisions between leavers and remainers.

The row about Big Ben’s bongs has raged all week. The House of Commons commission, which organises the running of parliament, had said the bell was “unlikely” to sound at 11pm on 31 January, the moment Brexit officially happens, in part due to the estimated cost of £500,000 to make it happen.

Work needed would have included restoring the clock chime mechanism, testing it, building a temporary floor in the belfry to allow the work to take place and delaying other restoration work for several weeks.

The new Speaker of the Commons, Lindsay Hoyle, who chairs the commission, said, “you’re looking at £50,000 a bong”.

The cabinet will meet outside London on the week of 31 January, in a bid to highlight the government’s plans for “levelling-up” regions outside London – and throw the focus forward to the future.

Johnson will claim that the UK’s formal departure from the EU at the end of this month fulfils his election pledge to “get Brexit done”. But it also marks the start of a race against time to secure a free trade agreement with the EU27 in time for the end of the transition period, in December.

The withdrawal agreement bill, which incorporates the Brexit deal struck by Johnson in Brussels last November, has been making rapid progress through parliament since the Conservatives won an 80-seat majority at last month’s general election.