The chances of Theresa May reintroducing her "snooper's charter" communications data bill are practically zero in the wake of the Guardian's disclosures on the scale of internet surveillance, leading Tory and Labour civil liberties campaigners have said.

David Davis, a former contender for Conservative leadership, and Tom Watson, the Labour deputy chair, both said on Thursday they felt there had been a change in the atmosphere at Westminster compared with the "great rush" to legislate in the immediate aftermath of the Woolwich murder of Drummer Lee Rigby.

Both MPs said the disclosure of the mass harvesting of personal communications, including internet data, by the American National Security Agency and Britain's eavesdropping agency, GCHQ, had shown that the existing UK regulatory framework was completely ineffective.

Davis said in particular that GCHQ's Tempora operation, which harvests global phone and internet traffic by tapping into the transatlantic fibre-optic cables, had "put up a big red flag" indicating it was time to think again from scratch about the legal oversight arrangements.

He said it was necessary to look at ways of rewriting the Regulation of Investigatory Powers Act 2000, which sets out the legal oversight arrangements for the interception and surveillance of communications.

But the former shadow home secretary and staunch Eurosceptic also praised the efforts of Viviane Reding, the EU commissioner for justice, who wrote to the foreign secretary, William Hague, on Wednesday giving him until the end of the week to answer the charge that the fundamental rights of citizens across Europe were being flouted.

"I hope that Viviane Reding keeps up the pressure. This is the only time you will hear me say that the European Union might be the answer," said Davis.

Watson said he shared Davis's analysis of the poor prospects for the reintroduction of May's communications data bill, which would require internet and phone companies to store for up to 12 months data tracking everyone's use of email, phone and internet.

Both were speaking at a packed Commons meeting organised by the Open Rights Group on the disclosures on Prism, the US surveillance programme, and GCHQ's Tempora programme, based on documents from the American whistleblower, Edward Snowden.

The meeting heard from surveillance experts Caspar Bowden, a former chief privacy adviser to Microsoft, and solicitor/advocate, Simon McKay. Bowden said a huge debt was owed to Snowden, who had made the most important disclosures about surveillance for more than 25 years.

He said the disclosures had serious implications for the corporate and individual stampede towards the use of "cloud computing" storage, much of which was housed in the US. He said that there was a real danger now that Britain would be left in an exposed position, with the rest of Europe not willing to allow their data to be stored through the UK. "Keep your cloudbase close and local and keep it in your jurisdiction," he said, adding that encryption was very limited as a defence.

Bowden, who co-authored a report for the European parliament on surveillance and has commented on the proposed EU data protection directive, which has yet to come into force principally because of British opposition, said he had ensured that amendments to the directive giving protection to whistleblowers were tabled in February.

He had also argued for a warning "pop-up" to be required when data was being transferred outside the EU's borders.

The foreign secretary earlier this week defended Britain's intelligence-sharing relationship with the US, saying it operated within the rule of law and arguing that terrorists, criminals and foreign intelligence agencies plotted against it in secret

"So we cannot protect the people of our countries without devising some of the response to those threats in secret," he said.

• This article was amended on 28 June 2013 to correct the spelling of Caspar Bowden's first name. It was further amended on 7 October 2013 to clarify that Bowden did not work as an adviser to the EU on its new data protection directive and that, rather than securing an amendment giving protection to whistleblowers, he ensured that amendments were tabled.