The backlash against Brexit has strengthened with more than 2.5 million people signed a parliamentary petition calling for a second EU referendum.

Demand for a fresh vote was so high that it crashed the parliament.uk website on Friday as the petition was shared widely on social media and new signatures were added at a rate of more than 100,000 an hour.

By Saturday night, more than 150,000 people had signed another petition calling on the London mayor to declare the capital independent from the UK so that it could stay in the EU, with supporters tweeting under the hashtag #londependence.

Senior Labour figures also called for parliament to overturn the referendum result. David Lammy, the MP for Tottenham, said: “Wake up. We do not have to do this. We can stop this madness and bring this nightmare to an end through a vote in parliament. Our sovereign parliament needs to now vote on whether we should exit the EU.

“The referendum was an advisory, non-binding referendum. The leave campaign’s platform has already unravelled and some people wish they hadn’t voted to leave. Parliament now needs to decide whether we should go forward with Brexit and there should be a vote in parliament next week.

“Let us not destroy our economy on the basis of lies and the hubris of Boris Johnson.”

The petition for a second EU referendum was set up by William Oliver Healey on 25 May. He wrote: “We the undersigned call upon HM government to implement a rule that if the remain or leave vote is less than 60% based on a turnout less than 75%, there should be another referendum.”

A parliament.uk map showed support for a second referendum was strongest in urban constituencies including Bristol West, Hornsey and Wood Green, and Jeremy Corbyn’s Islington North constituency, where more than 10,000 people signed the petition.

The lowest appetite was in Scotland, where some constituencies showed fewer than 500 supporters, although these areas have sparser populations on the whole.

Parliament must consider all petitions that reach a threshold of 100,000 votes. Few, however, reach 200,000. The most popular previous petition, with 823,346 signatories asking the government to give meningitis B vaccines to all children. A petition calling for Donald Trump to be barred from the UK was signed by 586,935 people.

In London, a growing number of people have been backing a debate about the future of the capital, where 70% rejected Brexit. Within 24 hours the petition on Change.org had been signed by more than 120,000 people, who backed freelance writer James O’Malley’s call: “Declare London independent from the UK and apply to join the EU”.

Supporter John Gowers wrote: “In a typical year, London contributes £10bn to £20bn to the rest of the UK and receives nothing in return. London has no control over its borders while it remains a member of the UK.”

Miles Jackson tweeted:

I don't think there's been a political cause in my lifetime I feel more strongly about than #londependence. I'm not British. I am a Londoner — Miles (@TheMilesJackson) June 24, 2016

But people outside the capital derided the petition, underlining the split between the capital and the rest of the UK. Laura Hughes from East Lothian tweeted: