MPs have approved Boris Johnson’s Brexit deal in a key vote that paves the way for the UK to leave the EU on January 31.

The Commons passed the prime minister’s withdrawal agreement bill (WAB) at second reading by 358 votes to 234, a majority of 124, following the Tories’ landslide election victory last week.

As the result was read out Tory MPs cheered loudly and minister Nigel Adams was heard saying “back of the net” as he stood next to the Speaker’s chair.

Six Labour MPs – Sarah Champion, Rosie Cooper, Jon Cruddas, Emma Lewell-Buck, Grahame Morris and Toby Perkins – backed the bill, while another 32 did not vote. Some will have been “slipped”, meaning they were excused from the vote due to prior commitments.

Opening the debate on the bill, Johnson insisted the old Leave and Remain labels were now as “defunct” as Montagues and Capulets at the end of Romeo and Juliet.

But he faced stinging criticism for dropping a legal commitment from the legislation that a deal would be struck with the EU so child refugees in Europe could continue to be reunited with their families in the UK after Brexit.

The PM claimed he was “absolutely committed” to allowing unaccompanied child refugees to be reunited with their families after Brexit.

But Labour leader Jeremy Corbyn said: “Coming to up to Christmas, shame on this government for abandoning children in this way.”

Labour MP Lisa Nandy – a candidate to replace Corbyn next year – said that while Johnson had won a mandate to get Brexit done, he had “not earned the right to shoehorn into this legislation measures that are a direct attack on some of the most vulnerable children in the world”.

Outside the Commons, Labour peer Alf Dubs, who put forward the original child refugees amendment, said it was “appalling and deeply distressing” that it had been removed.

Johnson has also used his majority to drop from the bill several other concessions made to opposition MPs.

These include protections for workers’ rights, MPs getting a veto on any decision to extend the post-Brexit transition period, and the Commons having a say in the government’s negotiating mandate.

The bill means Johnson’s deal, which settles issues including citizens’ rights, the £39bn “divorce” bill, and the Irish border, can be ratified in the UK. 

He is expected bring it back before MPs on January 7 so it can complete its passage through parliament in time for the UK to leave the EU and enter a year-long transition period, during which negotiations on a long-term future relationship will take place.

The bill also now outlaws any attempt to extend the transition beyond the end of December 2020, sparking concerns from Plaid Cymru MP Jonathan Edwards that Johnson has “boxed himself into a corner” by creating a new cliff-edge, and possible no deal, date.

Johnson replied: “On the contrary, I think most people looking at the negotiations would agree that it strengthens our negotiating position.

“If we have learned anything […] from the experience of the last three years, it is that drift and dither mean more acrimony and anguish.”

The PM appealed to the nation to come together and allow the “warmth and natural affection that we all share” for the UK’s European neighbours to “find renewed expression in one great new national project”.

“This is the time when we move on and discard the old labels of Leave and Remain,” he said.

“In fact, the very words seem tired to me – as defunct as Big-enders and Little-enders, or Montagues and Capulets at the end of the play.

“Now is the time to act together as one reinvigorated nation, one United Kingdom, filled with renewed confidence in our national destiny and determined at last to take advantage of the opportunities that now lie before us.”

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European Parliament vice president Pedro Silva Pereira said officials expect to conclude the EU ratification process by January 29.

He told BBC Radio 4′s Today programme: “We’ve always respected the choice of the British people, but it is true that it was a very long process.”

Asked what kind of trade deal can be negotiated and how “deep” it can be if it is to be completely concluded by the end of 2020, he said: “We have a very short timeframe available.

“Eleven months to negotiate such a complex trade agreement is unprecedented. It is a different situation. We come from a level of economic integration which has no comparison with other trade agreements that we’ve done before.”

He added: “The key issue will be what kind of regulatory disalignment we will have. The political declaration that we’ve agreed with the UK envisaged a very ambitious trading relationship with zero tariffs, zero quotas.

“But this can only be achieved if we ensure some regulatory alignment.”