Mr Allingham served with the Royal Naval Air Service during the Great War, later transferring to the Royal Air Force and serving at Ypres.
"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, East Sussex.
Mr Leader said: "He was very active right up to his final days, having recently celebrated his 113th birthday on HMS President surrounded by family.
"As well as possessing a great spirit of fun, he represented the last of a generation who gave a very great deal for us.
"Henry made many friends among the residents and staff at St Dunstan's. He was a great character and will be missed."
A funeral will take place later this month at St Nicholas' Church in Brighton.
Mr Allingham had five grandchildren and 12 great-grandchildren, 14 great-great grandchildren and one great-great-great grandchild.
He joined the Royal Naval Air Service in September 1915 before transferring to the RAF in April 1918 and was the last survivor of the Battle of Jutland.
Veterans Minister Kevan Jones said: "I am greatly saddened to hear of the death of Henry Allingham.
"For one of his age, his vigour for life was extraordinary.
"I was humbled to meet somebody who had led such an amazing life and we owe such a huge debt of gratitude to him and his generation. My thoughts are with his family."
Mr Allingham left a legacy of memories to the nation, one of his friends said today.
Dennis Goodwin, from the First World War Veterans' Association, said: "He left quite a legacy to the nation of memories of what it was like to have been in the First World War."
Mr Goodwin told BBC News that Mr Allingham's longevity could be put down to his being "essentially his own man".
Helen Emmerson, the manager of the care home where he spent his last years, said he had been determined to tell the world about his experiences despite his age.
She said: "I think particularly over the last six to nine months definitely he was finding it more and more difficult due to his advancing age.
"He had such strength and such strength of character. He wanted to get out there and spread the word."
She told Sky News: "He was tireless in his effort to get the word out there."
Mr Allingham became the world's oldest man in June.
Guinness World Records confirmed his title after the previous holder, Tomoji Tanabe, died aged 113.
Mr Goodwin, who was a close friend of Mr Allingham, said he was one of an "extremely unique and special generation of people".
"Not only did they survive the most horrific war of humanity but they had a new life to begin afterwards in an era of depression, and they did it admirably.
"I'm one of the products of that generation and I think my generation and other generation afterwards should remember that; it's a legacy they should create and keep in their memories."
Mr Goodwin, who was a regular visitor to Mr Allingham at St Dunstan's, described him as "an exceptionally good friend".
He added: "He was almost a surrogate father to me, we shared life together."
Describing the last few months of his life, he said: "He wasn't in the best of health. He wasn't eating.
"He often said to me he'd love to eat something good and have a drink like the old times but his tastebuds had gone, and I think it really frustrated him."
Mr Allingham's death means that Harry Patch, 111, the last survivor of the First World War trenches, is now Britain's oldest man. It is not yet known if he also now takes on Mr Allingham's title of world's oldest man.
Mr Patch, known as the Last Tommy, is Britain's last living soldier to have fought in the mud-soaked battle of Passchendaele in 1917 in which more than 70,000 British troops died.
He served as Lewis gunner with the 7th Duke of Cornwall's Light Infantry and now lives at a residential home in Somerset.
His recipe for a long life is in marked contrast to Mr Allingham's, who once put his longevity down to "cigarettes, whisky, and wild, wild women".
Mr Patch often said: "I neither smoke, drink nor gamble. Three sins - leave them alone."
A third known Great War survivor, Claude Choules, 108, who served with the Royal Navy, now lives in Australia. He born in Worcestershire.
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