The National Union of Teachers said the money was spent on 62 free schools, university technical colleges and studio schools which either closed, partially closed or did not open.
The NUT said the data was mainly drawn from government websites.
The Department for Education said free schools were popular with parents.
The figures were published by the National Union of Teachers during its annual conference - which had highlighted concerns about school budget shortages.
The union's general secretary Kevin Courtney said ministers should apologise to teachers and parents for the £138.5m "thrown away" on these abandoned projects.
"These figures make clear that the free school, UTC and studio school programmes were ill-thought policies which, in many cases, resulted in an appalling waste of significant sums of money," said Mr Courtney.
University technical colleges (UTCs) and studio schools have an emphasis on vocational skills for 14 to 19-year-olds.
"In the case of the closed UTCs, an average of £10m was spent on each school, rising to £15m in the case of Tottenham UTC.
"That sums of this magnitude have been thrown away at a time when schools across the country are crying out for funding for staff, to provide a broad and balanced curriculum and to ensure essential resources and equipment are available, is criminal."
Education Secretary Justine Greening this week announced that 131 new schools had been approved under the free school programme, creating around 69,000 places.
Free schools are state schools that are outside of local authority networks and which are set up by groups such as academy trusts, charities, parents and community groups.
There are now about 800 free schools either open or in the pipeline.
The NUT said the £138.5m on the closed or unopened schools would have paid for 3,680 teachers for a year.
Mr Courtney added: "The true cost of these policy failures is even greater. There is a human cost in the disruption caused to the education of the thousands of those pupils who attended schools which have closed.
"Usually it is local authorities who have had to pick up the pieces by finding alternative places for the displaced children.
"The NUT's biggest concern is that the government is intent on proceeding with these programmes despite growing evidence that the UTC and studio schools programmes cannot attract sufficient numbers of pupils."
The Department for Education said free schools were popular with parents and their creation had given parents more choices in finding good local schools.
"The construction costs of a newly-built free school are 29% lower than those built under the previous school building programme," said an education department spokeswoman.
"They also operate under a much more robust accountability system than council-run schools, meaning we can take swift action to deal with underperformance."
But Shadow Education Secretary Angela Rayner said the data was further evidence that the "free schools programme is a deeply inefficient way to provide the new school places that are desperately needed".
"Billions of pounds of taxpayers' money has been sunk into free schools, but there is still little evidence that the government is creating new places for children in the areas that they are most needed.
"Ministers should start ensuring that taxpayers' money is not being wasted and keep their promise to protect the funding that follows every child."

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Canada's Denis Shapovalov struck the ball in anger and hit Gabas during his game with Kyle Edmund, defaulting the match to hand the tie to Great Britain.
The International Tennis Federation said the 17-year-old's actions were "unintentional" but  fined him £5,600.
The ITF says the surgery was successful and Gabas is now recovering at home.
The Davis Cup World Group first-round tie in Ottawa was poised at 2-2, with Edmund leading 6-3 6-4 2-1 in the final match, but Canada's hopes ended when Shapovalov was disqualified.
He later apologised to Frenchman Gabas in the match referee's office and said he was "incredibly ashamed and embarrassed".
Gabas went to Ottawa General Hospital as a precaution, but no damage to the cornea or retina was found. On his return to France he had an X-ray, which revealed a fracture of the orbital bone under his left eye.
"I am now concentrating on rest and rehabilitation so I can be back on court in the near future," said Gabas.
Almost £140m has been "wasted" on free schools and other new types of school, which either closed early or failed to open at all, says a teachers' union.
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Umpire Arnaud Gabas has had surgery to repair a fractured eye socket after he was hit with a ball during Britain's Davis Cup win over Canada last week.