Just over a third of schools (35%) opened on the terms expected by the prime minister on 1 June, a union has found.
Research by the National Education Union (NEU) found 44% of schools did not open more widely to the any of the year groups, although they had remained open to look after key worker and vulnerable children.
The union also found clear regional variation in the proportion of schools meeting the government's criteria. Schools in the North East and North West of England were least likely to have opened to all eligible groups, with these regions having amongst the highest levels of Coronavirus.
Kevin Courtney, joint general secretary of the NEU, said: 'It was always reckless of Boris Johnson to set an arbitrary date and expect schools to fall in line.
'Heads and their staff know far more about their individual challenges than Whitehall ever will. As the regional variation according to coronavirus levels show, schools are listening to the science rather than politicians.'
The survey also found that schools face uncertainty around staffing, with 15% of schools asking staff who are clinically vulnerable to work from 1 June.
In a statement to parliament, education minister Gavin Williamson, said: 'We continue to follow the best scientific advice and believe that this cautious, phased return is the most sensible course of action to take.
'As the prime minister has set out, the Government will continue to monitor the rate of transmission carefully and will not hesitate to reintroduce restrictions on a local or regional basis if required.