However, the Met now says an investigation has been unable to prove intent to disrupt the event.
“This evening all six have had their bail cancelled and no further action will be taken,” the Met said in a statement.
“We regret that those six people arrested were unable to join the wider group of protesters in Trafalgar Square and elsewhere on the procession route.”
Officers arrested 64 people on coronation day, with 46 of those later bailed after being detained on suspicion of causing a public nuisance or breaching the peace.
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1:35
Republic chief on coronation arrests
Mr Smith said on Monday night that the Republic protesters who were arrested in London have all been told no charges will be brought against them.
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He also called for a “full inquiry” into who authorised the arrests during the “disgraceful episode”.
He said: “The speed with which they did this demonstrates they were very quickly aware they had made a very serious error of judgment and there will be action taken again.
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“I’m obviously relieved they dropped it so quickly but very angry they even went down this road, robbing people of their liberty for absolutely no reason.
“There was no evidence of any ability or intent to commit any offence and they simply decided to arrest us and that is outrageous.”
He added that a chief inspector and two other officers from the Met apologised to him personally at his home in Reading on Monday evening.
“I had three officers at my door personally apologising and handing the straps back to me. They were a chief inspector and two other officers from the Met. They seemed rather embarrassed to be honest,” he said.
“I said for the record I won’t accept the apology. We have a lot of questions to answer and we will be taking action.”
Prime Minister Rishi Sunak has backed the Met over arrests amid concerns they were cracking down on dissent on Saturday at the behest of politicians.
Mr Smith previously described the arrest of protesters during the coronation as a “direct attack on democracy” which showed that the right to peacefully demonstrate “no longer exists”.
In a tweet on Monday evening, he said: “We have just been told that the police will be taking no further action.
“This has been a disgraceful episode and we will be speaking to lawyers about taking legal action.
“I also expect a full inquiry into why they repeatedly lied to us and who authorised the arrests.”
Among the group’s members who felt “targeted” and silenced were nine people that held up signs reading “Not my King” at the Mall, moments before the procession left Buckingham Palace.
Although not arrested, they were whisked out of sight of the King and Queen into St James’s Park to be searched by Welsh police officers, outnumbering them roughly two-to-one.
Officers surrounded them and exchanged words before rifling through their pockets – some protesters faced the wall with their hands up in front of them.
Protester Harvey Woolf said: “I wasn’t very happy about the searches, I think we had been targeted because they didn’t want our message to get out.”
He continued: “What we are annoyed and disappointed about is that it was timed exactly to coincide with the point at which the carriage went down the Mall.”
The 66-year-old said police told him the protesters were suspected of carrying paint, but an officer told Sky News they led the Republic members away to avoid a “hostile environment” created by the crowds.
Royal supporters had booed and shouted “shame on you” when the protesters were initially marched away from the Mall.
The Met and Welsh police were asked if it was policy to remove people who are subject to a “hostile environment”, rather than those creating it, but were not able to immediately respond to a request for comment.
The protesters had been standing still holding bright yellow signs above their heads, several rows back from the barricades lining the Mall, to object to what they called hereditary privilege and power.
Republic were not the only group at the centre of a dispute over police behaviour on Saturday, with Westminster Council volunteers handing out rape alarms reportedlyarrested.
The arrests come in the wake of the Public Order Act, given royal assent on Tuesday, which handed the police more powers to curtail demonstrations, such as allowing officers to search people for items including locks and glue.
An ice alert for the South East and West Midlands is among new weather warnings now in force across the UK.
Wintry conditions caused widespread travel problems on Tuesday and forced more than 200 schools to close.
Forecasters say the difficult conditions will last until Wednesday morning, raising the prospect of a tricky rush hour on the roads and public transport.
The ice alert covers London and most of southern England as far west as Exeter, and as far north as Birmingham, Leicester and Chester, and expires at 10am.
A snow and ice warning also covers nearly all of Wales apart from some far western areas, most of western and northern Scotland and all of Northern Ireland – again until 10am.
Another snow and ice warning snakes up the coast from East Anglia to the Scottish border and is in force until noon.
The Met Office has warned some journeys by road and rail could take longer in the morning.
It says drivers should allow extra time and pack items such as a blanket, spade, phone charger, and food and water.
Walkers should also be wary of icy surfaces and cyclists are advised to stick to main roads that are more likely to have been treated.
Temperatures in many places are set to be close to or below zero overnight, while snow accumulations are forecast to range from a couple of centimetres to potentially 10cm or more on higher ground.
Areas above 300m in Scotland could see build-ups of 15-20cm, the Met Office said.
The Met Office’s chief meteorologist, Neil Armstrong, said: “With cold Arctic air firmly in place over the UK, continued winter hazards are likely through much of this week, with further updates to warnings likely in the coming days.”
Monday night saw sub-zero temperatures for much of the UK, reaching as low as -11.2C at Braemar, Aberdeenshire.
Snowfall was also widely reported, with 12cm recorded at Watnall in Nottinghamshire on Tuesday morning.
By Tuesday afternoon, around 140 school closures had been reported in Wales, around 50 in the West Midlands and 20 in Derbyshire.
There was also a travel headache for thousands of train passengers as National Rail warned of disruption to various routes.
By midday, around 64 of 120 services (53%) planned by East Midlands Railway had been cancelled or delayed by at least half an hour, according to punctuality website trains.im.
Figures for other operators included London North Eastern Railway (37%), Transport for Wales (13%), West Midlands Trains (13%) and CrossCountry (12%).
Rail travellers have been advised to check online before setting out and drivers should also plan ahead.
Darren Clark, from National Highways, said: “Gritters will be out treating our roads around the clock when ice or snow is forecast, but it is still important to drive to the conditions.
“Keep your distance and reduce your speed, because even in conditions that seem normal, and where the snow is not settling, it can be slippery if ice patches have formed, or where fresh grit has not been worked into the carriageway.
“Drivers should plan their journeys, monitor weather reports and pack a snow kit of blankets, food, water and a shovel.”
The first amber cold health alert of the season has also been issued by the UK Health Security Agency (UKHSA) covering all of England, except the South, until 6pm on Saturday.
The amber warning is issued when the weather is likely to “cause significant impacts across health and social care services”, including the potential for a rise in deaths, particularly among older people or those with health conditions.
Dr Agostinho Sousa, head of extreme events at the UKHSA, said: “It is vital to check in on vulnerable friends, family and neighbours to ensure they are well prepared for the onset of cold weather. Particularly if they are elderly or otherwise at increased risk.”
Yellow health alerts – the second lowest – are in place for the South East, South West and London.
More expected in coming days
The Met Office has warned there could be further heavy rain, strong wind and more snow during the rest of the week as temperatures stay below average
“Further snow accumulations are expected through the week, mostly by night at low levels, in northern Scotland and exposed parts elsewhere,” it said.
Those in the impacted areas have been told power cuts are possible and mobile phone coverage might be affected.
The Met Office said there was also a slight chance some rural communities could be cut off.
A woman whose body was found in the boot of a car last week died from strangulation, according to a post-mortem examination.
A manhunt is under way for Harshita Brella’s husband, Pankaj Lamba, who police believe has fled the country.
Officers said today that her body was discovered in a silver Vauxhall Corsa.
They believe she was killed on the evening of Sunday 10 November and driven in the early hours from the couple’s home in Corby, Northamptonshire, to Ilford in east London.
It’s understoodthe 24-year-old had been under the protection of a court order for victims of domestic abuse.
Northamptonshire Police said officers began searching for her after receiving a call with concerns for her welfare on 13 November – the day before she was found dead.
A post-mortem examination found Ms Brella’s preliminary cause of death was strangulation, police said.
New CCTV images of Pankaj Lamba, 23, have also been released as the manhunt continues.
Detective Chief Inspector Johnny Campbell, from the East Midlands major crime unit, said: “We suspect Harshita’s body had been placed in the boot of the vehicle by Lamba before leaving Corby.
“Lamba then abandoned the vehicle in Brisbane Road, Ilford, and made his getaway.”
Northamptonshire Police, the East Midlands major crime unit and the National Crime Agency are working together to try to find him.
An inquest into Ms Brella’s death will be opened on 20 November at The Guildhall in Northampton.
Anyone who thinks they might have seen Pankaj Lamba is urged to make contact as soon as possible.
“Any information, no matter how small, may be relevant to the investigation and help us get justice for Harshita,” said DCI Campbell.
People with relevant information can call 101, Crimestoppers anonymously on 0800 555111, or online at mipp.police.uk