The filmmaker - who was in his 80s - passed away after battling cancer, reports The Guardian.
He had been working on a new version of the classic Christmas film for its 30th anniversary.
Coates had recently been working with some of the film's original animators in London, but later became too ill to travel from Kent.
Channel 4 has confirmed that the new version will be broadcast this Christmas.
John Coates worked with The Snowman's original animator Raymond Briggs after reading the book. He convinced Channel 4 to make it as one of the channel's first films in 1981.
He was a former British army officer who served in World War II, and later worked as a film distributor in Asia.
Coates also worked on Beatles film Yellow Submarine, The Wind in the Willows and The World of Peter Rabbit and Friends.
The Snowman co-producer Camilla Deakin of Lupus Film said that he had seen the finished version of the new edition.
"He was happy to see it, comfortable and very, very happy with it," she said.
Speaking last year about the project, he said: "After 30 years, it's silly - why not have another one?
"We are hoping to repeat the success. We could fall flat on our faces. But it's a lovely story."
Watch the 'Walking in the Air' section from The Snowman below: