The Metropolitan Police said that two of the officers injured needed hospital treatment.
But things turned violent when police ordered the crowd to disperse, warning protesters they were violating the very social distancing and mask-wearing rules they had come to demonstrate against.
Police had to resort to using batons against some people as they had bottles hurled at them.
Commander Ade Adelekan, who is leading the Met operation, previously said that while there is "great frustration" at the regulations they were created to keep people safe from the virus.
"Likewise, there appeared to be no efforts by organisers to engage with crowds and keep those assembling safe from transmitting the virus". Protest is one of those exemptions, however organisers must submit a risk assessment and must comply with social distancing.
The Met said on Twitter: "Ten people have been arrested following demonstrations in Trafalgar Square, which moved to Hyde Park". In the interest of public safety, officers then worked quickly to disperse crowds, he said. I am grateful to those members of the demonstration who listened to officers and went home.
Earlier, police searched a man and confiscated a makeshift riot shield he was carrying.
Among the Saturday crowd was David Icke, a prominent conspiracy theorist who has drawn attention in the past by suggesting the world is secretly governed by reptilian humanoids, including the British royal family.
Mayor of London Sadiq Khan accused protesters of "reckless and violent behaviour" towards police.
"I urge all protestors to leave now. By flagrantly gathering in large numbers and ignoring social distancing, you are putting your health and the health of your loved ones at risk", he said.
The debate around acceptable government control during the pandemic has heightened as parliament prepares to review its COVID-19 legislation and the government considers further restrictions to control the disease. Some lawmakers have criticised the government for implementing the rules without parliamentary approval.
Police and protesters were injured when officers moved in after three hours of speeches in front of thousands of people who packed the iconic square in London waving placards opposing mandatory vaccinations, face mask requirements and limits on civil liberties.
A number of highly trained public order officers as well as mounted officers were among the visible police presence throughout central London, in order to respond "swiftly" to breaches of the coronavirus regulations and scenes of violence, the Met Police said.