Queen Elizabeth II and Prime Minister Gordon Brown led the tributes to the veteran, who symbolized the stoicism of the last generation of servicemen who saw the horrors of the Great War.
Allingham spoke of his experiences in the 1914-1918 conflict in order to remember fallen comrades shorn of the chance to live as long has he did and hoped there would be "no more wars".
In moving scenes last November, the wheelchair-bound Allingham tried for minutes to lay his wreath himself as he and two of Britain's three other surviving World War I veterans led the country in marking the 90th anniversary of the armistice.
Allingham spent his 113th birthday on June 6 at a party hosted by the Royal Navy aboard the frigate HMS President.
He become the world's oldest man on June 17, Guinness World Records confirmed, when the previous holder, Tomoji Tanabe of Japan, died aged 113.
"The Queen was saddened to hear of the death of Henry Allingham," a Buckingham Palace spokeswoman said.
"He was one of the generation who sacrificed so much for us all.
"Her thoughts are with his family during this time."
With his sight and hearing failing, Allingham spent his last years in a nursing home on the southern English coast.
"Everybody at St Dunstan's is saddened by Henry's loss and our sympathy goes out to his family," said Robert Leader, chief executive of St Dunstan's care home in Ovingdean, near Brighton.
Leader said Allingham was "very active right up to his final days."
"As well as possessing a great spirit of fun, he represented the last of a generation who gave a very great deal for us," he said.
"He was a great character and will be missed."
A funeral will take place later this month in Brighton.
"I had the privilege of meeting Henry many times," the prime minister said.
"He was a tremendous character, one of the last representatives of a generation of tremendous characters," he said.
"My thoughts are with his family as they mourn his passing but celebrate his life."
Allingham had five grandchildren and 12 great-grandchildren, 14 great-great grandchildren and one great-great-great grandchild.
Dorothy Allingham, his wife of 51 years, died in 1970, a decade after he had retired. In civilian life, he worked for car manufacturer Ford.
Born in Clapton, northeast London, Allingham witnessed three different centuries and saw six British monarchs on the throne.
Events he lived through included the death of queen Victoria in 1901, the sinking of the Titanic in 1912, the invention of television by John Logie Baird in the 1920s and the Wall Street crash of 1929.
A mechanic in the Royal Naval Air Service, he took part in the naval Battle of Jutland in 1916 and was one of the founding members of the Royal Air Force, which was formed in 1918.
With Allingham's death the only British World War I survivors are 110-year-old Harry Patch and 107-year-old Claude Choules, who lives in Australia.
Another veteran, William Stone, died in January at the age of 108.
The world's oldest man is now a 112-year-old American, Walter Breuning, who was born on September 21, 1896.
The world's oldest woman is Gertrude Baines, a 115-year-old American.
Prince Charles, who wrote the foreword to Allingham's 2008 autobiography "Kitchener's Last Volunteer", was saddened to hear of the veteran's death, a spokesman said.
Charles, the heir to the throne, had described Allingham as "one of our nation's historic treasures" in his foreword.
"He does not want modern society to forget what his generation gave for our futures but, equally, the message of peace and reconciliation is one that he desires to convey above all else," the prince wrote.
"We should all be humbled by this quiet, genial man and his desire to extol peace and friendship to the world, despite all the horrors he witnessed at such a young and impressionable age."
