Former Prime Minister Baroness Thatcher is "greatly saddened" by the death of Augusto Pinochet, said a spokesman.
Chile's former military leader, who has died aged 91 in hospital, backed the UK during the Falklands conflict.
Baroness Thatcher also pressed for his release after his arrest in London in 1998 over alleged human rights abuses.
Foreign Secretary Margaret Beckett used the death to pay tribute to Chile's "remarkable progress" since General Pinochet left office.
Mrs Beckett said Chile had become "an open, stable and prosperous democracy" since 1990.
A spokesman for Baroness Thatcher said the former PM would not be issuing a formal statement but would be sending "deepest condolences" to Gen Pinochet's widow and family.
The former military leader died in a Santiago hospital a week after suffering a heart attack.
During the Falklands conflict in 1982, Chile had been Britain's discreet ally against Argentina, and, despite Labour protests, Margaret Thatcher's government ended a ban on arms exports.
A regular visitor to the UK, where he had many friends, Gen Pinochet was arrested in London in October 1998, as he arrived to undergo medical treatment.
The Spanish government sought to put him on trial in Madrid over the deaths of its citizens in Chile.
Baroness Thatcher had tea with him and publicly expressed her opposition to the request.
Gen Pinochet remained under house arrest on the exclusive Wentworth Estate in Surrey for nearly 17 months before then Home Secretary Jack Straw decided he was too ill to stand trial.
Responding to Gen Pinochet's death, Mr Straw defended the decision by saying he acted in accordance with "clear legal duties".
"What I had to do throughout that 16 months of intense controversy about every decision was to apply the law as I saw it," he said.
"I was presented with evidence by the Chilean embassy of his mental and physical incapacity. I didn't accept that - instead I appointed four outside medical experts who, I was told, were amongst the toughest and most experienced of forensic medical experts in the country.
"They presented me with clear and unanimous conclusions that Pinochet would be unfit to stand trial in the UK. In those circumstances I had no option but to release him."
He said that when Pinochet appeared to recover on his homecoming, Mr Straw wondered whether he - and the four medical experts - had been "duped".
The former Conservative chancellor Lord Lamont, who opposed Pinochet's arrest, said Pinochet had had human rights issues to answer, but he had believed he should answer them in Chile - not Britain or Spain.
Lord Lamont opposed arrest on the basis that the Chilean government felt it would be an interference in Chilean affairs.
He also said arresting a head of state would have set a precedent "that would be very wrong".
"Above all, I felt this was a very difficult issue which Britain and Spain were not in a position to judge," Lord Lamont said.
Briton Dr Sheila Cassidy, who was tortured under the Gen Pinochet regime in the 1970s, told the BBC the victims would be disappointed he never faced a trial.
"For people who were tortured and particularly for people who've lost someone, whose daughter or lover was killed, I think that never goes behind you, and I think there will be a lot of anger that he has escaped prosecution," she said.
UK-based human rights groups have also been reacting to Gen Pinochet's death.
The decision not to extradite Pinochet attracted protests in 2000
More than 3,000 people were killed or "disappeared" during his leadership.
Amnesty International said Gen Pinochet's death was a "wake-up call" to governments that "speedy justice for human rights crimes" was needed.
A spokeswoman said his death must "not be the end of story".
"Amnesty International urges the Chilean authorities to declare the amnesty law void and proceed with investigations and prosecutions of all those others involved in the thousands of cases of `disappearances', torture and execution during Pinochet's period of rule," she said.
"Families and survivors need to know what happened, need justice and need their day in court."
The Medical Foundation for the Care of Victims of Torture had presented more than 60 cases in support of Gen Pinochet's 1998 extradition case on behalf of victims of the regime who had sought refuge in the UK.
"The fact that Augusto Pinochet will now not face a court of law to answer for the murderous activities of his regime is regrettable," said the charity's Leanne MacMillan.