The mayor of London, Sadiq Khan, said extra police had been deployed to reassure communities, especially those observing Ramadan, describing the attack as "an assault on all our shared values of tolerance, freedom and respect".
The attack comes at a time of political turmoil, as Prime Minister May plunges into divorce talks with the European Union weakened by the loss of her parliamentary majority in a June 8 election.
She has faced heavy criticism for her response to a fire in a London tower block on Wednesday which killed at least 58 people, and for her record on security after a series of attacks blamed on Islamist militants in recent months.
Miqdaad Versi, assistant secretary general of the Muslim Council of Britain, said the van had deliberately swerved into a group of people who were helping a man who was ill and had fallen to the ground.
The Finsbury Park Mosque gained notoriety more than a decade ago for sermons by radical cleric Abu Hamza al-Masri, who was sentenced to life in a U.S. prison in January 2015 for his conviction on terrorism-related charges.