AMNA NAWAZ: During a visit to Kenya, Britain's King Charles has stopped short of apologizing for his nation's repression of independence fighters 70 years ago.

But the new monarch is under severe pressure because of Britain's imperial past.

Caribbean nations are calling on Charles to dig into his $2 billion personal fortune and pay compensation for the royal family's slave-trading past.

Special correspondent Malcolm Brabant reports.

MALCOLM BRABANT: Britain's imperial state crown that Charles III wore at his coronation is supposed to represent the king's moral authority over his subjects in the United Kingdom, as well as former colonies which have retained the monarch as their head of state.

But, as Charles strives to bolster the monarchy's relevance in the 21st century, he's been undermined by revelations that the symbol of his sovereignty is stained by the blood of slaves.

ROBERT BECKFORD, University of Winchester: The British royal family are deeply entangled in the transatlantic chattel slave trade.

MALCOLM BRABANT: Professor Robert Beckford's field is social justice, and he's been investigating links between major institutions and slavery.

ROBERT BECKFORD: They signed the first charter that allowed privateers to go into Africa, into the West Indies, and traffic hundreds of thousands of enslaved people.

They made huge profits from it.

MALCOLM BRABANT: A document recently emerged showing that, in 1689, King William III accepted shares worth $300,000 in today's values in the slave-trading Royal African Company.

The donor was Edward Colston, the company's deputy governor, who made a fortune from trafficking 80,000 Africans to the Americans.

Three years ago, anti-racism activists tore down Colston's statue in his home city of Bristol.

Faced with growing proof of the crown's ties to slavery, King Charles has promised to support researchers by opening up the royal family's archives.

ROBERT BECKFORD: This provides them with a redemptive moment, an opportunity to do what no royal family has ever done before in the history of Britain, to acknowledge that much of their wealth is linked to the trafficking, enslavement and genocide of African people, and the opportunity to apologize for that and to pay reparation.

This could be a huge turning point, not just only in terms of British history, but in terms of world history.

MALCOLM BRABANT: One year after his accession, King Charles is under increasing pressure to apologize and pay reparations to Caribbean islands, which generated huge wealth from slave plantations and are now impoverished.

ARLEY GILL, Grenada National Reparations Commission: We want to encourage the royal family.

We are interested in all of the institutions, governments, families that have benefited.

They must come forward.

MALCOLM BRABANT: Arley Gill heads the authority seeking reparations to the island of Grenada.

Another major institution targeted by Gill is Lloyd's of London, the insurance exchange which profited from indemnifying the slave fleets.

ARLEY GILL: The reparation is not charity.

It is actually making amends.

MALCOLM BRABANT: According to the United Kingdom's National Archives, British ships transported roughly three million slaves across the Atlantic Ocean before the trade was outlawed in the early 19th century.

A study commissioned by the American Society for International Law, together with the University of the West Indies, calculates that Britain's slave debt amounts to $23 trillion.

The Caribbean islands are going after British institutions with slavery connections because the U.K. government is refusing to engage.

BELL RIBEIRO-ADDY, Labor Party Lawmaker: So, I want to ask the prime minister today if he would offer a full and meaningful apology for our country's role in slavery and colonialism and commit to reparatory justice.

RISHI SUNAK, British Prime Minister: No, Mr. Speaker, what I think our focus should now be on doing is, of course, understanding our history in all its parts, not running away from it, but right now making sure that we have a society which is inclusive and tolerant of people from all backgrounds.

But trying to unpick our history is not the right way forward and it's not something that we will focus our energies on.

ARLEY GILL: Rishi Sunak is on the wrong side of history.

And that must be made absolutely clear.

MALCOLM BRABANT: Dealing with Britain's colonial past is a tightrope walk for King Charles.

In Nairobi last night, he expressed remorse for Britain's brutality towards Kenyans during an insurgency in the 1950s.

KING CHARLES III, United Kingdom: The wrongdoings of the past are a cause of the greatest sorrow and the deepest regret.

There were abhorrent and unjustifiable acts of violence committed against Kenyans as they waged, as you said at the United Nations, a painful struggle for independence and sovereignty.

And, for that, there can be no excuse.

MALCOLM BRABANT: But the king stopped short of issuing an apology, which begs the question, just how far will he go when it comes to slavery?

The owners of this relatively modest dwelling in South England are trying to lead by example.

Retired Dr. Tom Trevelyan is a descendant of merchants who owned more than 1,000 slaves in Grenada and lived in this mansion.

TOM TREVELYAN, Reparations Campaigner: I did say to myself I have done nothing.

It's not that I'm apologizing for something I have done.

I'm apologizing for something that my ancestors did, because of the difference that it makes to the people who have been harmed.

MALCOLM BRABANT: Earlier this year Trevelyan and other members of his family went to Grenada to apologize in person.

His wife, Anita, works with other families trying to atone for the misdeeds of their ancestors.

ANITA TREVELYAN, Reparations Campaigner: Everybody in this country who's lived here has in one way or another benefited from the prosperity that the slave trade brought to this country, which enabled everything to be built on.

And it's just important to realize that, without that dreadful trade -- I mean, we have all eaten sugar.

We have all eaten chocolate.

We have all got cotton.

Where's it come from in the beginning?

LAURA TREVELYAN, Former BBC Foreign Correspondent: We're going to give it.

MALCOLM BRABANT: The Trevelyans' niece Laura, the former BBC foreign correspondent, has gone further by donating $120,000 to establish an education fund in Grenada.

Are there families who are reluctant to follow your path because they're afraid that they're going to lose their wealth?

LAURA TREVELYAN: Absolutely.

And that was a debate that happened within my family.

I mean, within mine, there isn't worth to lose really.

There is money and good middle-class lifestyles, for sure, but not hundreds of millions of pounds.

People who have that kind of generational wealth, I think, are concerned about the consequences, the consequences of being sued.

MALCOLM BRABANT: Most Jamaicans are descended from one million slaves who made fortunes for British plantation owners.

Their island, the second poorest nation in the Americas, is demanding compensation from Britain and is on track to become a republic by dumping Prince William's father as head of state.

PROTESTER: Respect us, man.

Apologize now.

PROTESTER: Reparations time come now.

PROTESTER: We will rebel!

This is living hell!

NICK DAVIS, One One Cacao: It's one of these terms which you see banded around a lot.

I think it's called generational trauma.

MALCOLM BRABANT: After being a journalist in Britain, Nick Davis is carving out a second career in Jamaica, making artisan chocolate.

NICK DAVIS: You only have to look at the murder rates of societies in the Caribbean.

It's a thing which was done to us which has been passed on from generation to generation.

We are literally killing ourselves, and that is because of this constant lack of resources, lack of opportunity.

MALCOLM BRABANT: Do you think reparations, if they were to happen in your lifetime, would make any difference to you?

NICK DAVIS: It's a tricky one.

I think that, as a community, we don't realize that what happened during the period of enslavement was our Holocaust.

When people tell you need to get on with it, it's in the past, realize how traumatic that is and how damaging that is to your very being.

MALCOLM BRABANT: Do you think that he personally should dig into his personal fortune and pay reparations?

ROBERT BECKFORD: That's the only way, as an economic entity, is to dig into your own profits from this genocide and make recompense.

And I'd expect King Charles to do just that.

MALCOLM BRABANT: So, the Caribbean is hoping Charles will follow the lead of other British slave-owning families, because it will pile pressure on the U.K. government to do the same.

For the "PBS NewsHour," I'm Malcolm Brabant in London.