Pyongyang's official news agency said Jang Song Thaek, 67, had been shot as a traitor for plotting to seize supreme power from his nephew.
Jang had previously been portrayed as a mentor to 30-year-old Kim - who inherited the position of supreme leader on the death of his father in 2011 - and was seen as the second most powerful figure in the secretive communist state.
But just days ago, he was shown being taken from a party event by uniformed guards, and North Korean media carried statements denouncing him as a "traitor to the nation for all ages", "worse than a dog" and "despicable human scum" who had been planning a military coup.
Prime Minister David Cameron's official spokesman told reporters: "If this is confirmed, it is another example of the extreme brutality of the North Korean regime."
His words echoed the comments of the White House, which also saw Jang's abrupt downfall and execution as evidence of the Pyongyang regime's "extreme brutality".
Foreign Office Minister Hugo Swire said: "We are deeply concerned to learn of the execution of Jang Song Thaek.
"This is another example of the brutality of the North Korean government and we have consistently raised concerns about severe and systematic human rights abuses in the country." Lord Alton of Liverpool, who chairs parliament's All-Party Parliamentary Group North Korea, said Jang had been viewed as a "real hope for reform".
He said: "I saw him at a distance at an official gathering after he'd been rehabilitated in 2006 and from the people I spoke to who knew him he represented for many the real hope for reform in North Korea.
"Jang Song Thaek's execution is a vivid and brutal reminder of the inherent and cruel nature of a regime that has always modelled itself on Stalin's USSR and the gulags."