In the clip, the journalist – who was wearing a BBC lanyard – was confronted by an unmasked mob near Downing Street shouting “traitor” and other slurs close to his face.
He was then forced to turn around and run towards a line of police officers as some of the group chased him.
The footage has been condemned by fellow journalists and MPs, including the prime minister and home secretary, Priti Patel.
Boris Johnson called the scenes “disgraceful”, tweeting: ”Disgraceful to see the hounding of Nick Watt doing his job.
“The media must be able to report the facts without fear or favour – they are the lifeblood of our democracy.”
Priti Patel also commented, tweeting: “The video of @BBCNewsnight’s Nick Watt being abused by a mob is appalling and distressing,” she tweeted. “This behaviour is never acceptable.”
Patel added: “The safety of journalists is fundamental to our democracy. This month the government launched a consultation to better understand the nature & volume of threats and abuse against journalists who are operating in the UK.”
The BBC said in a statement: “This behaviour is completely unacceptable.
“All journalists should be able to carry out their work without intimidation or impediment.”
Many others also tweeted their support for Nicholas and condemned the behaviour of the protesters.
Crowds had gathered in Westminster to protest against the government’s extension of coronavirus restrictions in England by four weeks.
The Metropolitan Police issued a statement, saying they are aware of the video.
They said: “The behaviour shown in the video is unacceptable. Members of the public, of any profession, have the right to go about their day without being subjected to verbal harassment or actions that put them in fear for their safety.
“In this instance, while officers were nearby as part of the policing response to the ongoing protest, they were not in the immediate vicinity of the incident.
“It was not clear at the time exactly what had taken place but after reviewing the video footage, a number of possible offences have been identified and an investigation has been launched.”
Anyone with information is asked to call 101 or Crimestoppers on 0800 555 111.