Long-awaited plans for better railways across the north of England have been given government backing with an undertaking to “reverse years of chronic underinvestment” by spending up to £45bn building Northern Powerhouse Rail.

Just over £1bn has been allocated to work up a detailed three-stage plan to connect cities from Liverpool to Newcastle, which could fulfil most of the demands of northern leaders, in a series of long-term projects.

However, mayors may have to raise local funding to pay for parts of the scheme, with the Treasury imposing a £45bn cap as it seeks to avoid the huge overspend that has blighted the construction of HS2.

The government also “set out its intention” to build a Birmingham-Manchester line after the completion of Northern Powerhouse Rail (NPR), although it insisted this was “not a reinstatement of HS2”.

It said NPR would form the backbone of a wider growth plan, with faster, more frequent train services transforming connections between cities.

Northern mayors broadly welcomed the three-stage plan, starting with the current works upgrading TransPennine links, which will be extended to a new Bradford station.

A new line connecting Liverpool and Manchester will be in the second stage of works – on a route via Manchester airport and Warrington that follows part of the axed HS2 high-speed rail plan, allowing a possible future link to Birmingham.

The third stage will bring further improved connections across the Pennines between Manchester, Leeds, Bradford, Sheffield and York.

Work is expected to start in the 2030s but not be completed until at least 2045, with construction of any further new line between Manchester and Birmingham starting only after that.

Mayors and local authorities, however, may need to raise revenue to ensure the NPR scheme goes ahead, as London did with Crossrail, should it breach the £45bn envelope.

Keir Starmer said the investment was “proof we’re putting our money where our mouth is, working with local leaders to deliver the transport links that will help working people do what they need to in life”.

The prime minister said people in the region had “been let down by broken promises”, adding: “This cycle has to end. No more paying lip service to the potential of the north, but backing it to the hilt.”

Government officials said that plans for how local contributions would be funded were being developed but could include business rates, tourist taxes or borrowing against future revenues.

Doubt remains over a key demand of Andy Burnham, the Greater Manchester mayor, for an underground station at Manchester Piccadilly. Burnham has argued it is essential to allow through trains and improve capacity and to build a new line without years of disruption and demolition in the city centre.

The cost difference between an underground and overground Piccadilly is thought to run into several billion pounds.

Burnham’s support for the plans had been in doubt, and he hinted at his frustration with the process earlier on Tuesday. Speaking at an Institute for Government event, the Labour mayor complained of having to “fight endlessly” in an “attritional battle” with Whitehall departments, accusing them of resisting devolution.

However, he welcomed the news as “a significant step forward”, describing it as a “an ambitious vision for the north, firm commitment to Northern Powerhouse Rail and an openness to an underground station in Manchester city centre”.

He said Manchester would “work at pace” to prove the case for an underground station, as well as detailed designs for the line to Liverpool.

Steve Rotheram, mayor of the Liverpool city region, welcomed what he called “a genuinely strategic approach … not another empty slogan or back of a fag packet plan but real investment, delivered in a proper partnership with local leaders”.

Manchester airport, the UK’s biggest outside London, will have a new station on the line. Ken O’Toole, the airport’s managing director, said it was a “long overdue step towards the creation of a highly productive and internationally competitive northern growth corridor”.

Yorkshire may be the biggest immediate beneficiary from the announcements, with Bradford confirmed to get a new station as part of the first stage of upgrades across the Pennines.

In a joint statement, the three Yorkshire mayors, Oliver Coppard, Tracy Brabin and David Skaith, said there was now “a clear national focus on connecting Sheffield, Leeds, Bradford and York with a frequent, electrified service”.

Development work will also be taken forward on reopening the Leamside Line, a 21-mile route in County Durham, which was closed in 1964.

Henri Murison, chief executive of the Northern Powerhouse Partnership, said the package would “enable a single labour market more like that of London and the south-east … The potential of the north will be unlocked, giving us better-paid jobs and new homes.”

However, it is understood that talks between Whitehall and some mayors went right to the wire, with Burnham said to have been reluctant to endorse the proposals as recently as Friday.

Another mayoral source said months of negotiations between Whitehall and northern leaders had led to a point where most mayors were happy: “The starting point for this was all wrong. We were concerned that it had to be a significantly new network and what some of the officials – not the politicians – have been advised was just the cheapest. We’ve ended up with something that’s workable.”