The National Association of Headteachers called on Mr Gummer to clarify if and when the reorganisation, originally hoped to take effect in April next year, would now go ahead.
The National Governors Council said the delay made planning education in the affected areas more difficult. Opposition parties have branded the delay, which means the new councils cannot be set up until after the next general election, a "complete shambles".
The authorities, which were due to take over education and other services now run by county councils, are Windsor and Maidenhead, Bracknell Forest, Newbury, Reading, Slough and Wokingham, all in Berkshire, and Hereford, Nottingham, Plymouth, Southend and Torbay.
Mr Gummer said the reorganisation had been postponed "to take account of the resources available", and said the Government also needed time to consider representations on the Local Government Commission's final report.
The Isle of Wight was the first unitary authority to start work last April. A further 17 in the areas now covered by Avon, Cleveland and Humberside, plus the city of York, are due to go ahead this April, and 13 more are due to hold elections to the new authorities in May and begin work next year.
George Varnava, president of the NAHT, said: "The Secretary of State has a responsibility to schools to remove the uncertainty and put an end to this stop-go policy. He must clarify what their future is to be.
"Schools are under considerable pressure to have development plans which take them three years into the future. Going unitary means their relationship with the local authority changes dramatically, and now that this has been delayed it adds to the uncertainty."
Berkshire's reorganisation, in which six districts were to become unitary, has been challenged in the courts by the county council.
Ann Risman, chair of Berkshire County Council's education committee, welcomed the delay. "It's good news for us. Our education service would suffer very badly if it were chopped up into six bits."
Simon Goodenough, chair of the National Governors Council and of the Devon Association of Governors, said: "Delaying this reorganisation without a clear date is not at all helpful. We would like to know what is going to happen so that governors can get on with the business of clarifying the new shape of local education authorities."