Cookson, who was re-elected as President for a seventh successive year at November's National Council, was appointed as vice chairman.
"I have now held both these demanding honorary roles since the new board set about rebuilding the organisation at the end of 1996. It was never my intention to keep both roles so long, but circumstances have not been right for a change until now," said Cookson.
"Over recent months, I have been promoted in my own career, and my spare time is under great pressure. I have felt for some time that the expansion plans currently being developed for British Cycling will demand more input from the president and the chairman if we are to be successful. Thus the combined role is no longer appropriate and I am delighted that Steve has taken over the chairman's role. He has provided much experienced input to the board's activities over the past few years and I am sure that, as chairman, he will be able to play a strong part in helping take the organisation forward."
Meredith, 55, has been a BCF member since 970. He served as national councillor and treasurer of the old Coventry division, and has been a member of the executive board since 1998. He is currently treasurer of West Midlands Division and chairman of the Kenilworth Wheelers CC.
His used to be MD of Cape Engineering UK of Warwick but is now executive director with board-level responsibilities for business development.
He has been closely involved this year with the development of the four-year Strategic Plan for British Cycling.
"I am convinced that the plan we have now drawn up will lead to a significant level of growth for
British Cycling over the next few years. Internationally we are becoming the envy of many of the more traditional 'top' cycling nations," said Meredith.
"We must now focus on building a stronger domestic racing scene and attracting many more participants into cycling. I believe the plan takes us in that direction."
Meredith was re-elected to the board at National Council, having reached the end of his term-of-office, and will serve on the board for a further three years.