So the simple answer is; there was, technically, no slavery in the United Kingdom. It was illegal, and any slave who reached the UK was no longer a slave, as the state of slavery did not exist in UK law. (In reality of course, a small number of slaves were illegally bought and sold, just as happens today.)
However, I suspect that what you're asking is a bit more complex, because you're probably not referring to the United Kingdom of Great Britain and Northern Ireland, which was only formed in 1801. Before that, there were two countries; the Kingdom of Ireland and the Kingdom of Great Britain, and I assume you're also interested in slavery in Great Britain (I don't know anything about slavery in Ireland, other than it existed in the middle ages).

Technically slavery hadn't existed in Great Britain since it faded into serfdom during the Middle Ages (see later in my answer). But while there was no legal way to enslave someone, in reality slaves brought from other countries to Great Britain were still treated as slaves. By the late 1700s, there were about 15000 black slaves in Britain, mostly domestic servants.
Of course, many wealthy British merchants and aristocrats owned slaves and were deeply involved in the slave trade; but this "triangle trade" was not about bringing slaves to Britain. Britain, after all, was a very unsuitable climate for plantations. Rather, slaves would be sent to work in the Caribbean (in brutal conditions) where they would grow consumer goods like cotton and sugar, which would then be brought back to Britain.

In 1769, a Scottish man named Charles Stewart returned from his job in Boston, Massachusets, to Britain. He brought with him an African slave he'd purchased in Boston named James Somerset. In 1771, James escaped. He had himself baptised as a Christian.
In November 1771, James was recaptured by Charles, who put him on a ship to Jamaica, intending to sell him to a plantation.
James's three godparents went to court, arguing that Charles had kidnapped James, a man over whom he had no legal authority. They were backed by an abolitionist named Granville Sharp.

James's lawyers argued, essentially, that there was no law on the English statute book covering slavery, ergo it did not exist. Yes, Boston had laws about slavery, and it was legal for Charles to buy slaves in Boston. But English law said nothing about slavery. Therefore, in England, James was not a slave, and by grabbing him by force and putting him on a ship for sale in Jamaica, Charles was kidnapping him.

Charles's lawyers, on the other hand, essentially argued that property law was the most important, and by freeing James the judge would set a dangerous precedent requiring all slaves in Britain to be freed.

The case became very famous in the press, and both sides received a lot of donations. The judge, Lord Mansfield, encouraged the two sides to settle, but both sides refused.

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Finally, Lord Mansfield made his judgement.

"that though the statute of Charles II had abolished tenure so far, that no man could be a villein regerdane [sic], yet if he would acknowledge himself a villein engrossed in any Court of Record, he knew of no way by which he could be entitled to his freedom without the consent of his master.

...

The cause returned is, the slave absented himself, and departed from his master's service, and refused to return and serve him during his stay in England; whereupon, by his master's orders, he was put on board the ship by force, and there detained in secure custody, to be carried out of the kingdom and sold. So high an act of dominion must derive its authority, if any such it has, from the law of the kingdom where executed. A foreigner cannot be imprisoned here on the authority of any law existing in his own country: the power of a master over his servant is different in all countries, more or less limited or extensive; the exercise of it therefore must always be regulated by the laws of the place where exercised.

The state of slavery is of such a nature, that it is incapable of now being introduced by Courts of Justice upon mere reasoning or inferences from any principles, natural or political; it must take its rise from positive law; the origin of it can in no country or age be traced back to any other source: immemorial usage preserves the memory of positive law long after all traces of the occasion; reason, authority, and time of its introduction are lost; and in a case so odious as the condition of slaves must be taken strictly, the power claimed by this return was never in use here; no master ever was allowed here to take a slave by force to be sold abroad because he had deserted from his service, or for any other reason whatever; we cannot say the cause set forth by this return is allowed or approved of by the laws of this kingdom, therefore the black must be discharged. "

Essentially, the argument was that slavery is not a natural state, but one which must be created by law. And that, in fact, no law in England ever made slavery legal, or allowed a master to enslave someone. He does mention serfdom (villeins), but mentions that Charles II abolished serfdom. So, therefore, James was not a slave, and Charles had no legal right to him.
And in fact this did lead to many slaves being freed in England, but not all, as it was not well enforced until 1807, and other legal cases argued that this case had only stopped slave-owners removing slaves from Britain, not actually owning slaves.

However, you'll notice that this judgement had no effect whatsoever on British colonies, which had their own laws. Thus the fact that slavery was illegal in Britain was of no help to the legions of slaves toiling for British owners in America, Jamaica, etc.

But you may also notice that this happened in 1772, a few years before the American revolution. Some historians have argued that slave-owners in America feared that now Britain had abolished slavery, they might be forced to similarly abolish slavery, and this may have pushed them to support American independence. This is not entirely clear, however.