Chris McManus and Franco Lamolinara, who were abducted last year, died as the rescue bid failed on Thursday.
The UK prime minister said he gave the go-ahead because the men's lives were in "imminent and growing danger".
British SBS special forces were "first in" during the rescue bid at Sokoto, says BBC correspondent Gordon Corera.
Gunmen seized the two engineers in the town of Birnin Kebbi on 12 May 2011.
They worked for B Stabilini, an Italian construction firm, and were involved in the construction of a local headquarters for the Central Bank of Nigeria.
"A Downing Street spokesperson said that the UK had been in regular contact with the Italian authorities throughout the case," BBC correspondent Gordon Corera reported.
"The spokesperson said Rome was informed when the operation was under way and that the British Prime Minister rang spoke to his Italian counterpart after its conclusion."
But speaking on the BBC's Newsnight, Italian Senator Lucio Malan said the Italians were not satisfied and wanted to know why they had not been consulted.
"It is quite uncommon that a country that is involved is not informed before. Apparently it was a very difficult situation and it might have been the best decision but it is still to be explained why the Italian authorities haven't been informed," he said.
"The intelligence which led to the raid is believed to have come through from co-operation with the Nigerians, with a source identifying the house where the hostages were being kept," the BBC's Gordon Corera reported.
"Events moved fast; British officials talk of a collapsing time frame with fears that the window of opportunity for a rescue might close as the men were either moved or killed.
"As members of the British SBS entered the compound it's thought one kidnapper was shot but the hostages were then found dead. officials say all the early indications are that they were killed by their captors."
Mr Cameron said he and the Nigerian authorities had decided to go ahead with the rescue operation after receiving "credible information about [the men's] location.
"A window of opportunity arose to secure their release. We also had reason to believe that their lives were under imminent and growing danger," he said.
Italian Prime Minister Mario Monti's office said he had asked Nigerian President Goodluck Jonathan to provide as soon as possible a "detailed reconstruction" of the events.
Meanwhile, BBC correspondent Alan Johnston said one MP from the centre-left Democratic Party was now demanding to know why Italy was not involved in the decision to go ahead with the operation.
President Jonathan said that the men's captors had been seized and "would be made to face the full wrath of the law".
He described them as being from Boko Haram, a militant Islamist group that has carried out a number of attacks on police, politicians and clerics who oppose it.
The BBC's Gordon Corera said he understood the attack was most likely to have been the work of a splinter cell within Boko Haram, with possible links to al-Qaeda.
Relatives of Mr McManus, who reports say was from Oldham in Greater Manchester, released a statement saying they were "devastated" by his death but thanked those who had worked to try to free him.
"During this ordeal we have relied heavily on the support of our family and friends which has never waned and has enabled us to get through the most difficult of times," they said.
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