A Labour government would compensate shareholders by issuing government bonds. The shadow chancellor said Labour had taken legal advice, including ensuring pension funds with investments in BT are not left out of pocket.
As with other planned Labour nationalisations, he said MPs would set the value of the company at the time of it being taken into public ownership.
Mr McDonnell told the BBC's Laura Kuenssberg that Labour would add an extra £15bn to the government's existing £5bn broadband strategy, with the money to come from the party's proposed Green Transformation Fund.
Having already announced plans to nationalise water, rail and now broadband, Mr McDonnell said this latest plan was "the limit of our ambitions".
A new entity, British Broadband, would run the network, with maintenance - estimated to cost £230m a year - to be covered by the new tax on companies such as Apple and Google.
"We think they should pay their way and other countries are following suit," said Mr McDonnell.
Labour has not yet completed the final details of how the internet giant tax would work, saying it would be based "percentage wise'' on global profits and UK sales, raising potentially as much as £6bn.
Mr McDonnell said that if other broadband providers did not want to give access to British Broadband, then they would also be taken into public ownership.
Labour has costed its policy from a report produced by Frontier Economics in 2018, external, which was originally produced for the Department of Digital, Culture Media and Sport.
Broadband packages in the UK cost households an average of around £30 a month, according to comparison site Cable - which people would no longer have to pay under Labour's scheme.
The party claims it would "literally eliminate bills for millions of people across the UK".
According to a report from regulator Ofcom earlier this year, only 7% of the UK has access to full-fibre broadband.
The government hit its target to bring superfast broadband to 95% of homes by December 2017 - at a cost of £1.7bn - but the internet speeds are significantly lower than those of full-fibre.
BT chief executive Philip Jansen told BBC Radio 4's Today programme Labour had under-estimated the price of its pledge.
But he said he was happy to work with whoever wins the election to help build a digital Britain, although the process for implementing Labour's plan would not be "straightforward".
He added that the impact of any changes on BT pensioners, employees, shareholders - and the millions of investors via pension schemes - needed to be carefully thought through.
The company has disputed the cost of rolling out fibre broadband to every home and business, saying it would cost closer to £40bn than £20bn.
The BBC's business editor, Simon Jack, said Labour's proposal had caught BT "off-guard", as Mr McDonnell had said in July that he had no plans to nationalise the telecoms giant.
Following Labour's announcement, BT's share price initially fell by 3% before recovering slightly.
Julian David, chief executive of TechUK, which represents many UK tech firms, said: "These proposals would be a disaster for the telecoms sector and the customers that it serves.
By BBC journalist Phil Mercer
Super fast, reliable and affordable - that was the broadband promise made to every Australian, who have been assured access to a new system by 2020.
The National Broadband Network (NBN) is described as "one of the largest and most complex infrastructure initiatives undertaken in Australia."
More than 6 million premises are already connected, but the quality of service is a lottery.
Some customers to the state-owned company are delighted, while others complain the network is sluggish.
There's a good reason for the disparity and it's down to the way that broadband is delivered. Some methods are much quicker than others.
In 2009, the former Labor government in Australia pledged an NBN with efficient optical fibre cables direct to most homes and businesses.
The current centre-right administration opted for a cheaper option that would be finished sooner, and chose instead a mix of technologies including optical fibre, copper wires, Hybrid Fibre Coaxial (HFC), fixed wireless and satellite.