Abolitionism was led by the Americans. The British tried to thwart abolitionist efforts.
https://en.wikipedia.org/wiki/Timeline_of_abolition_of_slavery_and_serfdom#1800%E2%80%931829
William Wilberforce's Slave Trade Act 1807 abolished the slave trade in the British Empire. It was not until the Slavery Abolition Act 1833 that the institution finally was abolished, but on a gradual basis. Since land owners in the British West Indies were losing their unpaid labourers, they received compensation totalling £20 million.The Royal Navy established the West Africa Squadron (or Preventative Squadron) at substantial expense in 1808 after Parliament passed the Slave Trade Act. The squadron's task was to suppress the Atlantic slave trade by patrolling the coast of West Africa, preventing the slave trade by force of arms, including the interception of slave ships from Europe, the United States, the Barbary pirates, West Africa and the Ottoman Empire.
That is a very broad brush.
The British slavers opposed it.
The rest of the British then ended slavery.
William Wilberforce led the way.
Once he flipped the British government on the issue - by getting a stealth bill passed that allowed the British Navy to seize ‘neutral’ flagged ships. It was all over.
That broke the power of the slavers in parliment. The new parliments then strongly opposed and banned slavery. The British navy then implemented the policy.
Just as in the US, there were slavers and free people.