The prime minister, who had met him a number of times, called Mr Allingham a "tremendous character".
Prince Charles reiterated his mother's comments, saying Mr Allingham belonged to that "incredible generation who did so much for their country".
A Clarence House spokesman said: "We owe him a huge debt of gratitude for all he has done."
Mr Allingham's grandson, Tim Gray, described his grandfather as very sincere and someone who loved life.
The 53-year-old, from Michigan in the United States, said: "He was always telling all these stories about the war, he wanted people to remember it.
"But he thought the ones that should really be remembered were the ones that died, not himself."
He added that "the whole family" would be coming to the UK from Canada and the US for his funeral, which takes place at St Nicholas' Church in Brighton later this month.
Dennis Goodwin, founder of the First World War Veterans' Association and a friend of Mr Allingham, said: "Henry was truly a gentleman - his strength of character, his purpose.
Air Chief Marshal Sir Glenn Torpy said he was an inspirational character
"He left quite a legacy to the nation of memories of what it was like to have been in WWI," he said.
For decades, Mr Allingham buried his memories of the war, avoiding reunions and refusing to tell his family about his experiences.
But, in recent years, he started making public appearances to make sure new generations did not forget the toll of war and went on to tell his life story in a book which was published last year.
First Sea Lord Admiral Sir Jonathon Band said: "Henry will be remembered with great fondness for his strong sense of humour and joy of life, and he was an inspiration for all those serving in the Royal Navy."
And Air Chief Marshal Sir Glenn Torpy, Chief of the Air Staff, said his passing was a "milestone in history".
"Henry was an inspiration to many and his thirst for life, cheery disposition and a desire to help others was his mark," he added.
The veteran's death has moved people across Britain.
Henry Allingham dies aged 113
At Lord's cricket ground in London, a minute's silence was held to mark his passing before play resumed on the third day of the second Ashes Test.
There were condolences too from staff at his Brighton care home, who said he had been very active right up to his final days and would be missed by all.
Last month, the Royal Navy hosted a 113th birthday party on HMS President in London for Mr Allingham and his family, close friends and members of the Royal Navy and Royal Air Force.
That was the last time his nephew Ronald Cator saw him.
Mr Cator said his uncle looked "very, very frail. He went downhill in the last few months".
"He used to sleep a lot at the end. He wanted to pass away, poor old boy."
His death means there are now only two British survivors of WWI - Harry Patch, 111, and Claude Choules, 108, now living in Australia.
Last month, Guinness World Records declared Mr Allingham, born in 1896, the world's oldest man following the death of another 113-year-old man in Japan.
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