Boris Johnson’s cabinet has given its approval for the whole of the HS2 high-speed train line to go ahead, with work expected to start within weeks.

The prime minister announced his decision to the House of Commons, saying the government believed it should proceed despite spiralling costs that could top £100bn.

He committed to both the London to Birmingham rail line and the next phase from Birmingham to Leeds and Manchester, saying he would integrate the project with Northern Powerhouse Rail, a wider scheme including a fast link between Leeds and Manchester.

However, he was critical of the management of the project, saying the company behind the project, HS2 Ltd, had “not made the task easier” and the costs “had exploded”.

Johnson said he would put a minister in charge of overseeing HS2 as a full-time job to avoid “further blowouts”.

HS2 Ltd will lose responsibility for redeveloping Euston station, which will be undertaken as a separate project, meaning high-speed trains are likely to terminate at Old Oak Common in London for some years.

A new body may also be set up to deliver the building of phase 2B – from Crewe to Manchester and the West Midlands to Leeds – along with Northern Powerhouse Rail, under the umbrella label of High Speed North. Johnson said the train line would go ahead in its entirety but there would be a review as it was “right that we interrogate the methods and the costs as we go forward”. That could mean that other parts of the overall network are built first.

“The cabinet has given high-speed rail the green signal. We are going to get this done,” Johnson said, adding that he hoped the first trains would run by the end of the decade.

The prime minister refused to commit to passing legislation for the second phase in this parliament, after questioning from the former transport committee chair Lilian Greenwood, which could alarm northern politicians fearing further delay.

Andy Burnham, the mayor of Greater Manchester, said he welcomed the headline commitment to HS2 but warned there “was a lot of detail lacking” on what would happen in the north.

“The prime minister has today listened and gone a considerable distance towards the case I made at the weekend for a new, integrated east-west and north-south railway for the north of England. That is why I welcome what he has announced today.”

But he asked the government to set a firm timetable for Northern Powerhouse Rail, as well as the northern HS2 spurs to Leeds and Manchester. He also demanded urgent upgrades to the existing railway, which he said “ruins journeys on a daily basis.”

Dan Jarvis, the mayor of the Sheffield city region, said the government must quickly complete its review of phase 2b, which includes a spur from Birmingham up through the east Midlands and Leeds via Sheffield.

“I’m acutely aware that some of our local residents now face further uncertainty and are unable to plan for the future,” he said.

He demanded the government use the budget in March to invest in “shovel-ready” projects in South Yorkshire and approve a £220m bid he has made to the Transforming Cities Fund for improvements to the local rail, bus, walking and cycling network.

“We also need clarity on the government’s plans to invest in buses so we can halt the decline in services in South Yorkshire,” he said.

The high-speed line’s growing price tag led Johnson to request a review of the scheme last summer. Douglas Oakervee’s assessment, published on Tuesday, “strongly advised against cancelling the scheme”, although it had many caveats on how to proceed, such as reducing the proposed number of trains per hour from 18 to 14 and reducing HS2 Ltd’s role.

Oakervee said the original rationale – a need for additional, reliable railway capacity – still held and there were no available alternatives. Upgrading existing lines would come at a high cost, including massive disruption for passengers, while cancellation would have “serious consequences for the supply chain, the fragile UK construction industry and confidence in UK infrastructure planning”.

The decision to go ahead will be controversial with many backbench Conservative MPs whose constituencies are affected by the train line running through them. However, there has been pressure on the government not to abandon HS2 from a number of regional politicians, including the West Midlands mayor, Andy Street.

Johnson also unveiled a host of other regional transport policies in an attempt to head off expected criticism over the money that will eventually be spent on HS2. He said: “The transport revolution is local. It must be local. But we cannot make these improvements in isolation from one another … We have to fix the spine.”

But some of his Tory backbenchers were deeply critical of the decision. Andrew Bridgen, the MP for North West Leicestershire, said it would be an “albatross round the neck” of the government and branded it “unloved, unwanted and grossly mismanaged”.

Environmental groups were also disappointed. John Sauven, the Greenpeace director, said it gave Johnson the “dubious honour of being this century’s largest destroyer of ancient woodlands in the UK”.

Jamie Peters, Friends of the Earth’s campaigns director, said: “HS2 is a costly and damaging mistake which will threaten wildlife, destroy ancient woodlands and do nothing to reduce climate-wrecking pollution.”

Emma Marsh, of the RSPB, said HS2 would destroy wildlife without “a viable plan for how it is going to mitigate or compensate for these losses”.

Jeremy Corbyn, the Labour leader, criticised the government’s management of the project, which was first announced by Labour in 2009, and called for fares to be affordable. “If it’s to have public support, the fares on HS2 must be affordable and comparable with the rest of the fare system on the railway network,” he said.

Corbyn highlighted the prime minister’s tendency to announce “big shiny projects” that later failed. Referring to No 10’s decision to explore the possibility of a bridge between Scotland and Northern Ireland, he joked: “Why not go the whole hog and make it a garden bridge connected to an airport on the sea? It stands as much chance of actually being built as any of those failed projects the former mayor of London put forward.”

Business groups and unions welcomed the HS2 decision. The British Chambers of Commerce director general, Adam Marshall, said it was “great news for businesses, investment and growth in many parts of the UK. It’s time to stop debating and start delivering.”

The Unite assistant general secretary, Gail Cartmail, said it was “good news for the economy in general and an immediate fillip to the construction sector”.