He was beaten to the top job at BZW in the summer when Barclays decided to hire Mr Harrison from Robert Fleming, the merchant bank, to succeed David Band, the chief executive who died at the beginning of the year.
BZW said Mr Brydon had "agreed to leave the firm as the new team takes over".
Mr Harrison has been on three months' gardening leave, and BZW said this had given him time to think about the way he wanted to run the business, which was why he was able to announce changes at the top so soon after arriving at his desk on Tuesday. One of Mr Harrison's key decisions has been to dispense with the post of deputy chief executive and instead ask the management committee to report directly to him as chief executive.
A BZW spokesman said: "With his experience at Lehman Brothers and Robert Fleming running investment banking businesses he knows the management structure needed. What he has done is bring in a new generation of people."
However, Mr Brydon, aged 50, is only three years older than Patrick O'Sullivan, who moves from chief executive of structured finance to chief operating officer and becomes a member of the management committee.
Mr Brydon made his name as a top fund manager at BZW, the investment bank set up in the mid-1980s, after joining Barclays' pension fund in 1977.
He became deputy chief executive of BZW and then acting chief executive for six months following the death of Mr Band. Mr Brydon is not thought to have taken any other job yet, but is likely to move swiftly into a top fund management job.
Mr Brydon said: "I am delighted that BZW is in such good shape and has a vigorous new management team to take it forward in the next stage of its development. It has been a privilege to have been a part of BZW's success and led it in the past few months."
Other key appointments to the management committee are Bob Diamond, who joined in July from CS First Boston as chief executive of global markets, and Steve Harker, who becomes chief executive of equities, replacing Jonathan Davie, who is to become full-time deputy chairman of BZW.
David Cannon, finance director and Sally Bott, head of human resources, also join the management committee.Reuse content