The three home associations, determined to safeguard their own national identities in world football, issued a joint statement rejecting what the BOA had hours earlier called a 'landmark' announcement.
In it, they repeated a collective opposition to their players joining any Team GB.
"We cannot support nor formally endorse the approach that has been proposed by the Football Association," said the joint statement.
"We have stressed this in communications to them and are disappointed that this has been ignored in the media release.
"No discussions took place with any of us, far less historic agreement been reached, prior to the statement from the BOA being released."
The BOA, the national Olympic Committee for Britain and Northern Ireland, had said in its statement that men's and women's football teams would be selected and managed by the English FA.
"Players from England, Scotland, Wales, Northern Ireland and other territories which fall under the BOA's remit as an NOC, who meet the approved competitive standard will be eligible for consideration and selection," it added.
A spokesman for the Welsh FA said separately that the BOA statement appeared to be "just a rehashing of old news and bits and pieces" and came "completely out of the blue".
"There is no new agreement signed or any deals reached," Ceri Stennett told Reuters. "The status quo remains the same."
The debate over the participation of players from Wales, Northern Ireland and Scotland in a British team has simmered since the Games were awarded to London in 2005.
The three countries compete with their own teams at international level but cannot do so at the Olympics, where only Britain is represented.
The three associations fear they will lose their independent status within football's world governing body FIFA if they allow their players to take part, despite their membership being enshrined in FIFA's statutes.
The BOA said the FA had consulted with the Scotland, Wales and Northern Ireland associations to develop the selection criteria and timeline.
"It has been a long, six-year journey to get to this point, with very real considerations from the Home Nations that first had to be recognised, respected and resolved," said BOA chief executive Andy Hunt.
"We absolutely respect the participation of the Home Nations as individual nations at all other football events," he added.
"I want to express my appreciation to all four Football Associations for their recognition of just how meaningful it will be for Team GB to compete in football in London 2012."
Scottish FA chief executive Stewart Regan suggested in an interview on the SFA website on Monday that the BOA's desire to present a united team might have more to do with selling tickets.
The second round of British ticket sales for London 2012 starts on Friday, with 1.7 million of the remaining 2.3 million tickets on offer being for the men's and women's footba
