Stocks in BP traded as low as 333 pence ($4.95) in London, the weakest since February 1997. The shares closed 4.4 percent lower at 334.20 pence.
Hayward is still overseeing the response to the spill but he is gradually transferring responsibility to Dudley, a 54-year-old American who lost out to Hayward for the CEO position three years ago, BP PLC spokesman Jon Pack said.
"The transition is happening right now," Pack said Tuesday.
Dudley will head the ongoing operation as well as the longer-term cleanup in the region affected by the spill. He will still report to Hayward, who BP said needed to focus on running the company.
As the new point man for BP's response to the disaster, Dudley could provide a fresh face after what have been views as public relations mistakes by Hayward. The British CEO was widely criticized in the U.S. for saying "I'd like my life back" and going yachting off the coast of England over the weekend while oil spill relief workers sweated in the Gulf of Mexico.
Dudley is experienced in protecting BP's interests under pressure. As chief of TNK-BP, a joint venture with a consortium of Russian billionaires, he steered the firm through a series of politically explosive disputes that saw one employee charged with espionage, the company's offices raided by Russian intelligence, an investor boycott and a barrage of tax and labor investigations.
BP had insisted earlier that Hayward would stay on the front line until the leaking well was plugged. The oil giant had forecast that the well will be capped in August.
But BP said Tuesday the transition will be complete sooner than that.