More than a dozen people are also injured from the incident after a truck careered down a cycle path near the 9/11 memorial site in Manhattan on Tuesday.
The driver, 29, was shot by police after jumping out of the rented vehicle with what turned out to be a pellet and paintball gun.
He was taken to a hospital and is in police custody.
Mrs May tweeted: “Appalled by this cowardly attack, my thoughts are with all affected. Together we will defeat the evil of terrorism. UK stands with New York City.”
Sadiq Khan also paid tribute, he said: “London stands in grief and solidarity with the great city of New York tonight after the despicable and cowardly terrorist attack in Manhattan.
“My heart goes out to the victims and their families.
“New Yorkers are strong and resilient – I know they will not be cowed by this assault on the innocent, and on our shared values and way of life.”
The Foreign Office said it is standing by to provide assistance to any Britons affected by the atrocity.
A spokesman said: “We are in contact with the local authorities and stand ready to provide any assistance but are not aware of any British citizens involved as yet.”
New York Mayor Bill de Blasio said the attack is “a particularly cowardly act of terror aimed at innocent civilians going about their daily lives”.
New York Governor Andrew Cuomo said it was a “lone wolf” attack and there is no evidence it is part of a wider plot.
Police said the vehicle, a rented Home Depot truck, entered the cycle path on West Street and mowed down several people.
The truck also smashed into a yellow school bus, injuring two adults and two children.
President Trump tweeted: “In NYC, looks like another attack by a very sick and deranged person. Law enforcement is following this closely. NOT IN THE USA!”
He later followed the tweet with: “We must not allow ISIS to return, or enter, our country after defeating them in the Middle East and elsewhere. Enough!”
It is not yet clear whether the attacker had any links to Islamic State.
He then added: “My thoughts, condolences and prayers to the victims and families of the New York City terrorist attack. God and your country are with you!”