Egyptian-born Badawi was the chairman of the Council of Mosques and Imams, founded the Muslim College in London and had been an imam of the London Central Mosque.
A spokeswoman for the college, where Badawi was the principal, said Badawi had been speaking at the Islamic Finance Summit in London when he was taken ill.
"He was at a conference at the time, he collapsed, an ambulance came and took him to hospital," she said.
His funeral would probably take place on Friday, she added.
Leading reformist Badawi forged close ties with the Christian and Jewish religions and joined with other faith leaders in a message of unity after the 7 July 2005 London bombings.
Tony Blair, the British prime minister, led a wave of tributes from political and religious leaders.
He described Badawi as a "wonderful mixture of the spiritual and the practical" who dedicated his life to the service of his faith.
Blair was speaking at the launch of the Christian Muslim Forum, a group tasked with promoting understanding between the two faiths, in London.
"It's most poignant that we should launch it on the day he most tragically died, for those of us who knew him would know what a special person he was," the British premier said.
Heir to the throne Prince Charles said: "The sudden loss of Zaki Badawi is a devastating blow to this country and to me personally.
"His brand of wisdom, scholarship, far-sightedness and above all humour has ensured that Zaki played an extraordinarily important role in the life of this country and amongst the Muslim community."
Badawi sparked controversy in the aftermath of the London bombings by suggesting Muslim women should remove the Islamic veil to avoid being targeted.
A week after the attacks, he was denied entry to the US with no explanation but accepted an unreserved apology offered later. He was due to speak in New York on the law and Islam.
Badawi was a frequent broadcaster and writer on Islamic matters.
Shami Chakrabarti, the director of human rights group Liberty, said: "People of all faiths, as well as those without faith, will mourn the loss of Dr Badawi, who provided unique leadership in a world gone mad."
"We pray that God almighty grants him a place in his paradise with the martyrs, the prophets and the righteous"
Sir Iqbal Sacranie,
General-Secretary, Muslim Council of Britain
Sir Iqbal Sacranie, general secretary of the main Muslim Council of Britain umbrella group, said Badawi's passing constituted a major loss for British Muslims.
"We pray that God almighty grants him a place in his paradise with the martyrs, the prophets and the righteous," he added.
Britain's Chief Rabbi Sir Jonathan Sacks said Badawi was "the face and voice of Islamic dignity and tolerance in Britain. He was a man of conscience and courage and I cherished his friendship."
The leader of the world's Anglicans, Archbishop of Canterbury Dr Rowan Williams, called Badawi a "uniquely effective interpreter of Islam" whose death was a great shock to those who worked with him in interfaith dialogue.