His close friend, comedian Jimmy Tarbuck, said: "He was one of the greats of British entertainment.
"I have nothing but lovely memories of him. He was a big, big influence on me.
He had the audience in the palm of his hand quicker than any other comedian I have seen.
"They loved him and you don't get that love very often.
Comedian Ken Dodd described Bygraves as a "magical performer"."He was absolutely brilliant - an excellent comedian, a very good singer and quite a good actor. Yes, he was a giant of showbusiness, a wonderful, wonderful man.
Veteran entertainer Des O'Connor said Bygraves had been a massive influence on his own career and had given him valuable guidance when he started out.
He said he wrote to him asking him for help and he was touched when he replied with a two-page handwritten letter full of advice.
O'Connor said: "I used to use it as a rule book when I was starting out and over the years everything he's said has been helpful to me."
He described Bygraves as a "top, top professional".
The couple had moved from their home in Poole, Dorset, to Australia several years ago to take advantage of the warm climate.Chirpy cockney Bygraves, one of nine children, lived in a council flat in Rotherhithe but went on to become a multi-millionaire.
He became famous for his catchphrase: "I wanna tell you a story" and recorded songs including You Need Hands and Tulips From Amsterdam.
Bygraves's agent Johnny Mans said he had been diagnosed with Alzheimer's two years ago.
"He had become confused and often wasn't sure where he was," he said.
"He was in good health otherwise but would have turned 90 on October 16 so was pretty ancient.
"We were hoping to do a big concert for him soon."
His son Anthony Bygraves said his father, who went on to host game show Family Fortunes, had gained lots of experience from entertaining during the war years when he did more than 1,000 shows.