Nutkins, who is survived by his wife and eight children, was being treated for leukaemia in hospital.
His agent John Miles confirmed the news in a statement, adding: "He had fought for about nine months or so with acute leukaemia.
"He was an absolutely lovely guy and just loved animals and he was never happier than when he was with animals. We will all miss him very, very much."
An English naturalist, television presenter and author, Nutkins found fame in the 1980s when he became a co-presenter of the BBC children's television series Animal Magic with Johnny Morris. During his time on the show, he gripped viewers with his relationship with Californian sea lion Gemini, who he hand-reared.
He went on to establish himself as a popular face on kids TV when he anchored The Really Wild Show alongside Nicola Davies, Chris Packham and Sue Dawson between 1986 and 1993.
Nutkins's passion for animals began at a young age when he worked at London Zoo, occasionally skipping school so he could spend more time with the elephants. He was also mentored by author Gavin Maxwell, who he helped look after otters with on the west coast of Scotland.
His love for animals was surprisingly not affected by an incident when he was aged 15, where he lost two fingers after being bitten by an otter called Edal.
Nutkins other TV credits included Brilliant Creatures, Growing Up Wild and most recently an appearance on the BBC's Winterwatch and BBC Three series My Life As An Animal.
His enthusiasm for animals, his quirky hairstyle and straight-talking approach endeared Nutkins to mainstream audiences. He wasn't afraid to criticise the decline of modern-day naturalist TV series, famously saying: "I have no time for people who bulls**t with animals! I'd see Bill Oddie on Springwatch and think, 'I could do a better bloody job than you, mate!' He's a twitcher, a Goodie - not a natural history expert."
The Countryside Alliance gave a tribute on Twitter, commenting: "He taught an entire generation to love the natural world and animals - education is so important."
Meanwhile, ex-children's TV presenter Phillip Schofield wrote: "So sad to hear of the death of Terry Nutkins. I worked with him often in my 'broom cupboard' days. A delightful man and passionate naturalist."
Comic Ricky Gervais said: "RIP Terry Nutkins. Animal lover and thoroughly nice chap."
Watch some clips of a documentary about Terry Nutkins: