London-born Bellamy, who became a household name as a TV personality, scientist and conservationist, died on Wednesday, according to the foundation.
His colleague, David Shreeve, described him as a "larger-than-life character" who "inspired a whole generation".
In later life Bellamy, who lived in County Durham, attracted criticism for dismissing global warming.
In 2004 he described it as "poppycock" - a stance which he later said cost him his TV career.
Bellamy worked in a sweet factory and as a plumber before embarking on his broadcasting career.
His scientific career began when he got a job in the biology department of a technical college in Surrey, he told BBC Radio 4's Desert Island Discs programme in 1978.
It was there that he met his future wife, Rosemary.
But it was on a trip to Scotland where he discovered his love for plants, he told the programme.
"I got really turned on by plants and I found out that if somebody told me what a plant was, I just couldn't forget it," he said.
He gained public recognition for his work as an environmental consultant over the Torrey Canyon oil spill, when a tanker was shipwrecked off the coast of Cornwall in 1967.
He went on to present programmes such as Don't Ask Me, Bellamy On Botany, Bellamy's Britain, Bellamy's Europe and Bellamy's Backyard Safari.
And in 1979 he won Bafta's Richard Dimbleby Award, for best presenter of factual programmes.
His distinctive voice also inspired comedian Sir Lenny Henry's catchphrase "grapple me grapenuts".
BBC arts correspondent David Sillito described Bellamy as "the enthusiastic face of botany on television" for more than 30 years.
In 2003, Bellamy told BBC News that he was sceptical about mankind being responsible for rising temperatures and suggested that they might be part of the Earth's natural cycles.
He said: "We have got to get this thing argued out in public properly and not just take one opinion."
Ten years later, he told the Independent newspaper: "It (global warming) is not happening at all, but if you get the idea that people's children will die because of CO2 they fall for it."