As well as Triesman, Gill and Mawhinney, an as yet unappointed chief executive and chief operating officer will sit on the board. Gerry Sutfliffe, the Minister for Sport, will represent the Government.
Sir Keith Mills, chief executive of London's successful bid to host the 2012 Olympics, Baroness Amos and Sir Martin Sorrelll will act as non-executive directors.
Richard Caborn, the former Minister for Sport, will attend meetings but have no voting rights.
Triesman said a World Cup in England in 10 years' time can have a lasting positive impact on the game.
"I am confident this team will put together a bid which exceeds Fifa's requirements and expectations by making the best possible case for a World Cup in England," he said.
"We also want to show that a World Cup on these shores will benefit the development of football around the world and create positive change.
"England has to earn the right to stage the 2018 World Cup. Over the coming months, we will have to show we can do so to the highest standards and with the right values.
"People around the world football family will have views about what produces a great World Cup and we will be listening and learning. If we have the honour of succeeding it will be because we earned the right from the members of our international family."