Thousands of people have marched through central London under a heavy police presence to protest against lockdown measures, and 33 were arrested.

Demonstrators gathered at Speakers’ Corner by Hyde Park at about midday where anti-lockdown figurehead Piers Corbyn gave a speech saying he would “never take a vaccine” and falsely claiming that the scale of deaths from Covid was not dissimilar to those from flu each year.

As police surrounded him and detained a handful people as they ordered demonstrators to disperse, the crowd then marched out of the park and through London from Marble Arch.

On Friday 62 MPs and peers wrote to the home secretary saying that allowing the police to criminalise people for protesting was “not acceptable and is arguably not lawful”, in a letter coordinated by Liberty and Big Brother Watch.

They said the right to protest was enshrined in human rights law, amid growing scrutiny of police tactics after officers forcibly dispersed demonstrators at a vigil for Sarah Everard last week.

The specific exemption to coronavirus regulations in England allowing the right to protest was removed in November, but some legal experts have said it remains a “reasonable excuse” for leaving home. It is widely accepted that transmission of coronavirus is far less likely outdoors.

The protest on Saturday afternoon passed peacefully, with police appearing to apprehend relatively small numbers of people as a helicopter hovered above, but at about 4pm officers began stepping in to separate crowds and continued to urge people to disperse, detaining some.

The scenes back at Speaker’s Corner after 5pm became increasingly fractious as officers ordered crowds to leave and arrested more protesters amid chants of “freedom”.

There were minor clashes as people objected to some heavy-handed policing and officers were pelted with bottles; they responded by raising their batons and leading many more demonstrators away in handcuffs to their vans.

At about 7pm a hundred police officers wearing riot helmets and carrying shields arrived at Hyde Park and urged people to go home.

Other protests were expected to take place elsewhere in the UK.

The Met tweeted that officers were “engaging with those gathering around Piccadilly and other areas of central London to protest, explaining that we remain in a public health crisis and urging people to disperse or go home.”

It said previously: “Current government legislation makes gatherings in groups of more than two people unlawful unless exemptions apply. Gathering for the purpose of a protest is not an exception under the Covid-19 regulations. The right to protest must be balanced against the rights of others and the protection of public health.”

Last week a high court judge suggested that the human rights of expression and gathering might be considered reasonable excuses in some circumstances, and the Met accepted there was no blanket ban on protest.

If people are not distancing, however, it may become illegal, according to the regulations. Campaigners say the police should work to facilitate Covid-safe demonstrations. Previous demonstrations have been broken up by police.

The letter to Priti Patel was signed by several Conservative MPs including Steve Baker and Christopher Chope, along with the Liberal Democrat leader, Ed Davey, and a number of Labour MPs and peers including Diane Abbott and Shami Chakrabarti.

It urged Patel to immediately “expressly exempt protests from restrictions on gatherings in all tier areas” and highlighted the current situation in which deeming the legality of a protest is the responsibility of the police on a case-by-case basis.

“There is no legal certainty for the police as regards their duties and powers, and no legal certainty for protesters as regards their rights. This is not acceptable and is arguably not lawful,” the letter said.

Sam Grant, the head of policy and campaigns at Liberty, said: “We must all be able to stand up to power and have our voices heard. In a healthy democracy, protest is a critical way we can fight for what we believe in. The government’s current quasi-ban on protest is completely unacceptable.”

Referring to the policing bill, which passed its second reading in the Commons this week despite opposition from more than 150 human rights charities, unions and faith communities, as well as Labour, Grant added: “Using short-term restrictions on protest to stifle dissent while they pass permanent ones is as absurd as it is authoritarian.”

Silkie Carlo, the director of Big Brother Watch, said: “A country cannot be described as a democracy if people do not have the freedom to protest. The harrowing scenes of police officers using force against women at Clapham Common recently were avoidable and wrong. Over the past week, many more demonstrators and even legal observers have been arrested or fined.”

But the Home Office has maintained, in comments to the BBC, that “while we are still in a pandemic we continue to urge people to avoid mass gatherings, in line with wider coronavirus restrictions”.