MPs are to vote within weeks on the creation of a London Olympics-style delivery body to oversee the multibillion-pound restoration of the Houses of Parliament, with work expected to begin in the mid-2020s.

Sources close to the project, which could cost up to £6bn and last until the mid-2030s, have suggested that opposition to the restoration plans had dramatically fallen since the devastating fire at Notre Dame Cathedral in Paris, which had prompted a new sense of urgency to move it forward.

Efforts to get on with restoring the Palace of Westminster have also been buoyed by the removal of Gavin Williamson as defence secretary. He had objected to the use of the Ministry of Defence (MoD) car park for contractors’ lorries on security grounds, meaning plans were being redrawn, causing lengthy delays.

It is estimated by sources familiar with the project that Williamson’s departure may have saved the taxpayer “hundreds of millions of pounds” and speed up the project considerably if contractors are allowed to use the car park.

MPs are expected to leave the Palace of Westminster in 2025 and move into a temporary chamber in the former Department of Health building Richmond House, which is next to the MoD. Lords are to move into the Queen Elizabeth II conference centre.

The government is expected to introduce the parliamentary buildings (restoration and renewal) bill on Wednesday and a second reading is expected within days, after MPs voted to go ahead with the restoration last year.

The bill proposes creating a sponsor body of MPs, peers and external experts who would act as the client on behalf of parliament for a delivery authority of experts to manage the complex project.

Andrea Leadsom, the leader of the House of Commons, said the fire at Notre Dame had brought home “the importance of preserving our historic buildings”. She said the bill would protect the future of the Palace of Westminster and ensure value for money.

Fire safety teams constantly patrol the neo-gothic Palace of Westminster, where there have been 66 fires since 2008. In 2016, the Guardian reported a malfunctioning light on an obscure part of the roof caused an electrical fire that could have spread rapidly had it not been detected.

Parliamentary opponents of the project, who feared spiralling costs and lengthy delays, were also said to have been convinced about the need for restoration by a series of serious masonry falls that could have killed passersby.

It is unclear whether MPs will manage to pass the full bill before the summer recess but a “shadow” sponsor body has already been created that will begin recruitment for the delivery body once second reading is passed.

Plans for the temporary Commons chamber in Richmond House are set to be unveiled on Wednesday, after experts determined that more ambitious plans such as a “floating parliament” on the Thames would be a security risk. MPs must also be able to access other parts of the parliamentary estate, such as offices in Portcullis House, without leaving the secure zones.

Under the plans, the Richmond House chamber, which will only have 75% of the capacity of the current chamber, will remain a permanent exhibition and educational space once MPs move back into the Palace of Westminster.